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Topics in the asymptotic analysis of linear and nonlinear eigenvalue problems Lindsay, Alan Euan 2010

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Topics in the asymptotic analysis of linear and nonlinear eigenvalue problems. by Alan Euan Lindsay A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) July 2010 c© Alan Euan Lindsay 2010 Abstract In Applied Mathematics, linear and nonlinear eigenvalue problems arise frequently when characterizing the equilibria of various physical systems. In this thesis, two specific problems are studied, the first of which has its roots in micro engineering and concerns Micro-Electro Mechanical Systems (MEMS). A MEMS device consists of an elastic beam deflecting in the presence of an electric field. Modelling such devices leads to nonlinear eigenvalue problems of second and fourth order whose solution properties are investigated by a variety of asymptotic and numerical techniques. The second problem studied in this thesis considers the optimal strategy for dis- tributing a fixed quantity of resources in a bounded two dimensional domain so as to minimize the probability of extinction of some species evolving in the domain. Math- ematically, this involves the study of an indefinite weight eigenvalue problem on an arbitrary two dimensional domain with homogeneous Neumann boundary conditions, and the optimization of the principal eigenvalue of this problem. Under the assumption that resources are placed on small patches whose area relative to that of the entire do- main is small, the underlying eigenvalue problem is solved explicitly using the method of matched asymptotic expansions and several important qualitative results are estab- lished. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Micro-Electro Mechanical Systems . . . . . . . . . . . . . . . . . . . . 2 1.2 Persistence on Patchy Domains - An Eigenvalue Optimization Problem 10 2 Mathematical Modeling of Micro-Electro Mechanical Systems . . . 16 2.1 Lumped Mass Spring Model . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 A Model From Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . 19 3 Fold Point Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1 Numerical Solution Nonlinear Eigenvalue Problems . . . . . . . . . . . 24 3.1.1 Simple Upper Bounds For λc . . . . . . . . . . . . . . . . . . . . 30 3.2 Biharmonic Nonlinear Eigenvalue Problem: Slab Geometry . . . . . . . 32 3.3 Biharmonic Nonlinear Eigenvalue Problem: Multidimensional Domain . 39 3.4 Perturbing from the Pure Biharmonic Eigenvalue Problem . . . . . . . 45 3.5 The Fringing-Field and Annulus Problems . . . . . . . . . . . . . . . . 47 3.5.1 Fringing-Field Problem . . . . . . . . . . . . . . . . . . . . . . . 47 3.5.2 The Annulus Problem . . . . . . . . . . . . . . . . . . . . . . . . 48 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4 Multiple Fold Points and Singular Asymptotics . . . . . . . . . . . . . 52 4.1 Asymptotics of the Infinite Fold Points Structure . . . . . . . . . . . . 53 4.1.1 Infinite Number of Fold Points For N = 1 . . . . . . . . . . . . . 53 4.1.2 Infinite Number of Fold Points For N > 1 . . . . . . . . . . . . . 56 iii Table of Contents 4.1.3 Fold Points of The Bratu Problem . . . . . . . . . . . . . . . . . 63 4.2 Asymptotics of The Maximal Solution Branch as λ→ 0: Slab Domain . 69 4.2.1 The Fringing Field Problem . . . . . . . . . . . . . . . . . . . . 69 4.2.2 The Beam Problem . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3 Asymptotics of Maximal Solution Branch as λ→ 0: Unit Disk . . . . . 85 4.3.1 The Beam Problem . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3.2 The Annulus Problem . . . . . . . . . . . . . . . . . . . . . . . . 99 4.4 Concentration Phenomena General Domains . . . . . . . . . . . . . . . 104 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5 Persistence in Patchy Domains . . . . . . . . . . . . . . . . . . . . . . . 116 5.1 Determination of The Persistence Threshold for One Patch . . . . . . . 118 5.1.1 A Single Interior Patch . . . . . . . . . . . . . . . . . . . . . . . 118 5.1.2 A Single Boundary Patch . . . . . . . . . . . . . . . . . . . . . . 124 5.2 The Persistence Threshold for Multiple Patches . . . . . . . . . . . . . 128 5.3 Effect of Habitat Fragmentation and Location on Species Persistence . 135 5.3.1 The Persistence Threshold for One Patch . . . . . . . . . . . . . 135 5.3.2 Multiple Patches and The Effect Of Fragmentation . . . . . . . 139 5.3.3 Optimization at Second Order . . . . . . . . . . . . . . . . . . . 148 5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 6.1 Micro-Electro Mechanical Systems . . . . . . . . . . . . . . . . . . . . . 156 6.1.1 Concentration in Other Nonlinear Eigenvalue Problems . . . . . 158 6.1.2 Quenching Behavior in Fourth Order Time-Dependent MEMS . 159 6.2 Eigenvalue Optimization Problems in Mathematical Ecology . . . . . . 162 6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 iv List of Tables 3.1 Upper Bounds on the Fold Point Location. . . . . . . . . . . . . . . . . 32 4.1 Numerical Values of A and φ. . . . . . . . . . . . . . . . . . . . . . . . . 58 v List of Figures 1.1 Schematic Plot of a MEMS Capacitor. . . . . . . . . . . . . . . . . . . . 3 1.2 Bifurcation Diagram From Lumped Mass-Spring Model. . . . . . . . . . 4 1.3 Bifurcation Diagram with Infinite Fold Points Structure. . . . . . . . . . 5 1.4 Schematic Plot of a Two-Dimensional Patchy Habitat. . . . . . . . . . . 13 2.1 Schematic diagram of lumped mass spring model of MEMS. . . . . . . . 16 2.2 Bifurcation diagram of lumped mass spring model. . . . . . . . . . . . . 18 3.1 Bifurcation diagram with the infinite fold points structure. . . . . . . . . 25 3.2 Bifurcation Diagrams of Beam Problem. . . . . . . . . . . . . . . . . . . 27 3.3 Finite Number of Fold Points in Perturbed Problems . . . . . . . . . . . 28 3.4 Bifurcation diagrams on Slab and Square . . . . . . . . . . . . . . . . . 30 3.5 Global bifurcation diagrams. . . . . . . . . . . . . . . . . . . . . . . . . 40 3.6 Global Bifurcation Diagrams. . . . . . . . . . . . . . . . . . . . . . . . . 45 3.7 Asymptotic and Numerical Fold Point Location . . . . . . . . . . . . . . 47 3.8 Asymptotic Fold Point Location. . . . . . . . . . . . . . . . . . . . . . . 49 4.1 Graph of Constants A and φ. . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Bifurcation Curves of Membrane MEMS . . . . . . . . . . . . . . . . . . 62 4.3 Asymptotic Bifurcation Diagrams for Membrane MEMS. . . . . . . . . . 63 4.4 Inner Behaviour of Bratu Problem. . . . . . . . . . . . . . . . . . . . . . 68 4.5 Numerical and Asymptotic Predictions for Bratu and N = 3. . . . . . . 68 4.6 Asymptotic Bifurcation Diagrams for Mixed Biharmonic Problem. . . . 73 4.7 Asymptotic Bifurcation Diagram for Biharmonic MEMS. . . . . . . . . 80 4.8 Asymptotic Bifurcation Diagrams for Mixed Biharmonic MEMS. . . . . 85 4.9 Asymptotic Bifurcations for Biharmonic MEMS. . . . . . . . . . . . . . 99 4.10 Asymptotic Bifurcation Diagram for Annulus. . . . . . . . . . . . . . . . 104 5.1 Principal Eigenvalue in Concentric Disks. . . . . . . . . . . . . . . . . . 124 5.2 Example of a Perturbed Disk . . . . . . . . . . . . . . . . . . . . . . . . 138 5.3 Numerical verification of perturbed Green’s function. . . . . . . . . . . . 139 5.4 Illustration of Qualitative Results. . . . . . . . . . . . . . . . . . . . . . 145 vi List of Figures 5.5 Illustration of Strategy With Pre-Existing Patches. . . . . . . . . . . . . 147 5.6 Illustration of Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.7 Example of Optimization of µ1. . . . . . . . . . . . . . . . . . . . . . . . 152 6.1 Multiple Touchdown on Unit Square. . . . . . . . . . . . . . . . . . . . . 160 6.2 Multiple Touchdown Points. . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.3 Time Dependent Touchdown. . . . . . . . . . . . . . . . . . . . . . . . . 162 6.4 Bifurcation Diagram Showing The Allee Effect. . . . . . . . . . . . . . . 164 vii Acknowledgements I would like to thank very much my supervisor Michael Ward whose expert guidance and passion for Mathematics has been inspirational. I would also like to thank Professors Neil Balmforth, Dan Coombs and Brian Wetton who have been greatly supportive throughout my studies. For making my time at UBC so enjoyable, there are many people whose deserve acknowledgement, but special thanks is certainly due to Sasha, Ryan, Sionnach, Steve, Kelp, Jason, Hana and Omer and all the Vaughn-Jones/Metcalf family. For always being there for me, thank you Mum, Dad and Fraser. I would also like to acknowledge the financial support of a NSERC postgraduate fellowship throughout my studies and the support of all the Mathematics office staff. viii Chapter 1 Introduction The method of matched asymptotic expansions has become a staple technique in the toolbox of applied mathematicians over the last 50 years. Its original and perhaps still most significant contribution, was the deduction of the boundary layer equations whereby, under an assumption of high Reynolds number, the well-known Navier-Stokes equations are decoupled into two simpler problems. One problem concerns a region of inviscid flow where the effects of viscosity are neglected and another region, governed by the boundary equations, where the no slip condition can be applied. In essence, the method approximated the solution of one intractable problem by two much simpler ones along with precise bounds on the error induced by this decoupling process. Subsequently, matched asymptotic methods, or more generally singular perturba- tion methods, have evolved into a vast array of techniques which can be employed to systematically reduce difficult problems arising in the natural sciences and engineering into a sequence of tractable ones. These reductions make use of a small ( or large ) parameter arising in the problem to make assertions about the relative importance of its components. This parameter typically appears naturally when considering physical problems, however, it may also be introduced to facilitate the analysis of a problem in a particular regime. Both of these scenarios are realized in this work. This thesis is concerned with an application of contemporary asymptotic and numer- ical methods to two particular problems in Applied Mathematics. The first problem has its origin in the field of Engineering and arises from modeling a class of devices called Micro-Electro Mechanical Systems (MEMS). The second problem has its roots in Ecol- ogy and addresses the question of deploying a fixed quantity of resource in a bounded two dimensional domain so as to ensure the survival of some species evolving in the domain for the largest range of physical parameters. Mathematically, this involves the study of a indefinite weight eigenvalue problem on an arbitrary two dimensional domain with homogeneous Neumann boundary conditions, and requires the optimization of the principal eigenvalue of this problem. The following sections introduce in detail the problems to be considered, provide the reader with an account of previous work undertaken and outline the aims of the thesis. 1 1.1. Micro-Electro Mechanical Systems 1.1 Micro-Electro Mechanical Systems In 1959, Nobel Laureate and Physicist Richard P. Feynman gave a lecture titled There’s Plenty of Room at the Bottom and initiated a new field of human endeavour. In his seminal lecture, he posed the question Why cannot we write the entire 24 volumes of the Encyclopedia Britannica on the head of a pin? He argued that to accomplish this, one would need to shrink the text by a factor of 25, 000 which equates, roughly, to writing with a pen whose point has an area corresponding to 1, 000 atoms - an achievable task. Feynman used this example to demonstrate that a huge quantity of information can be stored on small scales. Feynman went on to outline two additional avenues for exploration; the miniatur- ization of circuits and of motors. The natural progression of the former has led to the microchip; an invention which has subsequently revolutionized almost every aspect of science and technology. The natural progression of the latter has led to the development of Micro-Electro Mechanical Systems (MEMS) which are now on the verge of providing a similar revolution. A MEMS device is a combination of mechanical components and integrated circuits constructed on a miniature scale. Current manufacturing techniques have made it possible to construct fully functioning motors visible only with the aid of a microscope; accelerometers with length less than one millimeter; needles so tiny they administer injections without stimulating nerve cells and many more devices, all with numerous applications. However, these advances have not been matched by our understanding and ability to control physical processes on such small scales. Indeed, when shrunk by many orders of magnitude, a device whose mechanical pro- cesses may be well understood on a macro scale must be reexamined as different physical processes become dominant at small scales. For example, MEMS devices typically have a large surface area to volume ratio which favors electrostatic forces over magnetic forces as a means of actuation. The need to understand physical processes on such small scales has given mathematical modeling a central role in the design of MEMS devices. Simple capacitance devices consisting of a rigid inelastic conducting ground plate opposite a thin deformable elastic plate, held fixed along its boundary, are a key com- ponent of MEMS (c.f. Fig. 1.1). The upper part of this device consists of a thin deformable elastic plate that is held clamped along its boundary, and which lies above a fixed ground plate. When a voltage V is applied to the upper plate, the upper plate can exhibit a significant deflection towards the lower ground plate. Beyond some critical voltage V ∗ called the pull-in voltage, the deflecting plate can touchdown on the ground plate, an event which can compromise the operation of some devices and is essential for the operation of others, e.g. switches and valves. 2 1.1. Micro-Electro Mechanical Systems d Ω L Elastic plate at potential V Free or supported boundary Fixed ground plate y′ z′ x′ Figure 1.1: Schematic plot of the MEMS capacitor with a deformable elastic upper surface that deflects towards the fixed lower surface under an applied voltage. From a point of view of MEMS design, a larger value of V ∗ would be detrimental to devices which require touchdown to occur in their operation as this would necessitate a higher input voltage. On the other hand, transducer type devices which operate by measuring one quantity through its effect on the deflection of the upper plate, may require a larger operating voltage range to prevent touchdown. In every case accurate determination of the maximum voltage V ∗ is essential to effective MEMS design and is a central goal in mathematical modeling of the device’s operation. The first batch fabricated Micro Electro-Mechanical System (MEMS) device was the resonant gate transistor (RGT) created in 1964 at Westinghouse [4]. The device, manufactured on a length scale of 0.1mm, was essentially a tuning fork which provided a resonant output to an electrical input of a predetermined frequency. One of the key components of the RGT was the capacitor like device displayed in Fig. 1.1. The authors developed a simple model of the device based on the assumption that the deflection of the upper surface was a function of time only. By incorporating the mechanical properties of the device into a mass-spring system, the ordinary differential equation (ODE) α2 d2u dt2 + du dt + u = λ (1− u)2 . (1.1.1) was proposed for the dimensionless deflection u(t) of the upper plate. The dimen- sionless parameter α is known as the quality factor in engineering parlance while the non-negative parameter λ, proportional to V 2, represents the relative importance of electrostatic and mechanical forces in the system. The steps leading to (1.1.1) and dis- cussion of its predictions are presented in § 2.1. To investigate the pull-in instability, as predicted by (1.1.1), time derivatives are set to zero to determine that equilibrium 3 1.1. Micro-Electro Mechanical Systems deflections of the device satisfy the relationship u(1 − u)2 = λ. The solutions of this expression are conveniently displayed in the bifurcation diagram of Fig. 1.2. 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.03 0.06 0.09 0.12 0.15 u λ (λ∗, u∗) Figure 1.2: Bifurcation diagram of steady state solutions to equation (1.1.1). A saddle-node or fold bifurcation is observed at the point (λ∗, u∗) = (4/27, 1/3) with a stable (solid) and an unstable (dashed) solution branch emanating from the bifurcation point. The bifurcation diagram of Fig. 1.2 indicates that for λ < λ∗ = 4/27 one stable and one unstable equilibria are attained, while for λ > λ∗, no equilibrium deflections are possible. The critical value, λ∗, represents the non-dimensional pull-in voltage of the device and knowledge of its value is key to predicting the behaviour of the device. Despite the simple nature of this model, i.e. no elastic effects and no boundary conditions at the edges of the device, it does capture the pull-in instability and provides a compact expression for the pull-in voltage. In an effort to better capture the geometry and the material properties of the de- flecting surface, Pelesko coupled the theories of linear elasticity and electrostatics [33] (see also [59] and the references therein) to show that the dimensionless equilibrium deflection u(x) of the upper plate satisfies the partial differential equation (PDE) ∆u = λ (1 + u)2 , x ∈ Ω ; u = 0 x ∈ ∂Ω . (1.1.2) The non-negative quantity λ, which is directly proportional to the square of the voltage V applied to the upper plate, represents a ratio of electrostatic and elastic forces in the system and acts as a natural bifurcation parameter. A solution of (1.1.2) can be interpreted physically as a balance between an elastic restoring force and an attracting coulomb force. The model (1.1.2) was derived in [33] from a narrow-gap asymptotic analysis. The steps involved in the derivation of (1.1.2) are detailed in § 2.2. This simple nonlinear eigenvalue problem has been studied using formal asymptotic analysis in [34] and [17] for the unit slab Ω = (0, 1) and the unit disk Ω = {(x, y)|x2 + y2 ≤ 1}. For the unit disk, one of the key qualitative features for (1.1.2) is that the 4 1.1. Micro-Electro Mechanical Systems 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 |u(0)| λ (a) Ω = {|x| ≤ 1}. 0.4400 0.4425 0.4450 0.4475 0.4500 0.975 0.980 0.985 0.990 0.995 1.000 |u(0)| λ (b) Ω = {|x| ≤ 1} (Zoomed). Figure 1.3: Panel (a) shows steady state solutions for the unit disc Ω = {|x| ≤ 1} and is seen to exhibit multiple fold points as |u(0)| approaches 1 from below. The critical value is determined to be λ∗ = 0.789. Panel (b) is an enlargement of the upper portion of panel (a) and shows the curve undergoing additional folds as |u(0)| nears 1. bifurcation diagram |u(0)| versus λ for radially symmetric solutions of (1.1.2) has an infinite number of fold points with λ→ 4/9 as |u(0)| → 1 (cf. [34], Fig. 1.3). Analytical bounds for the pull-in voltage instability threshold, representing the fold point location λc at the end of the minimal 1 solution branch for (1.1.2), have been derived (cf. [34], [13], [17]). A generalization of (1.1.2) that has received considerable interest from a mathematical viewpoint a problem with a variable permittivity profile |x|α in an N -dimensional domain Ω, namely ∆u = λ|x|α (1 + u)2 , x ∈ Ω ; u = 0 x ∈ ∂Ω . (1.1.3) There are now many rigorous results for (1.1.2) and (1.1.3) in the unit ball in spatial dimension N and in more general domains Ω. In particular, upper and lower bounds for λc have been derived for (1.1.2) and for (1.1.3) for the range of parameters α and N where solution multiplicity occurs (cf. [13]). In [20] it has been proved that there are an infinite number of fold points for (1.1.2) in a certain class of symmetric domains. Many other rigorous results for solution multiplicity for (1.1.3) under various ranges of α and N have been obtained in [13], [9], and [19]. In this work, three distinct perturbations to (1.1.2) and their effect on the pull-in voltage and the infinite fold point structure are investigated. These are; 1A solution whose L2 norm is smaller than any other solution. Similarly a maximal solution has a larger L2 norm than any other solution. 5 1.1. Micro-Electro Mechanical Systems • The beam problem: − δ∆2u+∆u = λ (1 + u)2 , x ∈ Ω ; u = ∂nu = 0 x ∈ ∂Ω. (1.1.4) In this case the deflecting surface exhibits rigidity and δ represents the relative importance of tension and rigidity. • The fringing field problem: ∆u = λ(1 + δ|∇u|2) (1 + u)2 , x ∈ Ω ; u = 0 x ∈ ∂Ω. (1.1.5) Here, δ ≡ d2/L2 is the square of the aspect ratio and the term δ|∇u|2 accounts for the fact that the electric field between the plates is not quite uniform. • The annulus problem: ∆u = λ (1 + u)2 , δ < |x| < 1; u(δ) = u(1) = 0. (1.1.6) The value of δ > 0 represents the radius of a small circular sub-domain removed from the unit disc. Although the quantity δ has a different physical interpretation in each of the three above cases, it is used generally to represent a small perturbation from the unperturbed membrane problem (1.1.2). The bihamonic term −δ∆2u of (1.1.4) arises by considering the deflecting surface to be a plate supporting bending stresses. By contrast with (1.1.2) and (1.1.3), only a few rigorous results are available for (1.1.4) and similar fourth-order variants. Under Navier boundary conditions u = ∆u = 0 on ∂Ω, the existence of a maximal solution for (1.1.4) was proved in [26] and its uniqueness established in [22]. Under Navier boundary conditions and in the three-dimensional unit ball it, was proved in [21] that −∆2u = λ/(1 + u)2 has infinitely many fold points for the bifurcation branch corresponding to radially symmetric solutions. In [8], the regularity of the minimal solution branch together with bounds for the pull-in voltage for the corresponding clamped problem −∆2u = λ/(1 + u)2 with u = ∂nu = 0 on |x| = 1 in the N -dimensional unit ball are established for N ≤ 8. Some related rigorous results are given in [5]. The fringing fields problem (1.1.5) was developed in [38] from careful consideration of the edge effects in the electric field of a drum shaped device with disk-shaped membrane end sections. It was shown in [38] that such edge effects induce a global perturbation of the basic nonlinear eigenvalue problem, replacing λ with λ(1+δ|∇u|2) where δ = (d/L)2 6 1.1. Micro-Electro Mechanical Systems is the aspect ratio of a MEMS device with gap width between the upper and lower surfaces d and whose surfaces have length scale L as shown in Fig. 1.1. For the unit disk, equation (1.1.5) was studied numerically in § 5 of [38], where it was shown that the effect of the fringing-field is to reduce the pull-in voltage. In addition, numerical investigation of (1.1.5) indicated that the effect of the fringing-field is to destroy the infinite fold point structure of the basic membrane problem (1.1.2) in the unit disk, leaving a finite number of fold points on the upper branch of solutions followed by a branch of solutions with limiting behaviour λ→ 0 as |u(0)| → 1. The third modification of the membrane problem (1.1.2) in the unit disk is to pin the rim of a concentric inner disk in the undeflected state (cf. [35], [10]). The perturbed problem for 0 < δ ≪ 1 in the concentric circular domain δ < |x| < 1 is formulated as in (1.1.6). This change in the domain topology due to the insertion of a small inner disk has a two-fold effect on the solution. First, it increases significantly the pull-in voltage. Second, it allows for the existence of non-radially symmetric solutions that bifurcate off the radially symmetric solution branch (cf. [35], [10]). For δ ≪ 1, the problems (1.1.4), (1.1.5), and (1.1.6), can all be viewed as perturba- tions of the basic and well-studied membrane problem (1.1.2). Of principal interest is the effect of these perturbations on the pull-in stability and on the infinite fold point structure of the unperturbed problem (1.1.2). The work undertaken to investigate these points is now described. Chapter 2, describes the steps required to derive MEMS equations (1.1.1)-(1.1.6). In §3.1, the numerical methods employed to solve the relevant equations are outlined and preliminary findings discussed. Generally, it is observed that each of the aforementioned perturbations has a pronounced effect on the fold point location and that they also act to destroy the infinite fold points structure, leaving a solution branch with a finite number of turns. For the unit slab and unit disk, a simple upper bound for the fold point location λc at the end of the minimal solution branch for (1.1.4) is derived and calculated numerically. The remainder of Chapter 3 concentrates on the effect of the perturbations on fold point location. A rather precise determination of the pull-in voltage threshold is required for the actual design of a MEMS capacitor since, typically, the operating range of the device is chosen rather close to the pull-in instability threshold (cf. [33], [38]). Therefore, in mathematical terms, the primary goal of the analysis is to calculate asymptotic expansions for the fold point location λc at the end of the lower, or minimal, solution branch for (1.1.4), (1.1.5), and (1.1.6), in the limit δ → 0. In §3.3, asymptotic expansions for the fold point location of the biharmonic problem (1.1.4), denoted by λc, at the end of the minimal solution branch are derived in the limiting parameter ranges δ ≪ 1 and δ ≫ 1 for an arbitrary domain Ω with smooth 7 1.1. Micro-Electro Mechanical Systems boundary. To treat the δ ≪ 1 limit of (1.1.4), singular perturbation techniques are used to resolve the boundary layer near the boundary ∂Ω of Ω, which arises from the term δ∆2u in (1.1.4). This analysis yields effective boundary conditions for the corresponding outer solution, which is defined away from an O(δ1/2) neighbourhood of ∂Ω. Then, appropriate solvability conditions are imposed to determine analytical formulae for the coefficients in the asymptotic expansion of λc for δ ≪ 1. These coefficients are evaluated numerically for the unit slab and the unit disk. By contrast, the analysis of (1.1.4) for the limiting case δ ≫ 1 consists of a regular perturbation expansion of the solution to the pure biharmonic nonlinear eigenvalue problem −∆2u = λ̃/(1 + u)2, with λ̃ ≡ λ/δ. In this way, it is shown for the unit disk and the unit slab that λc ∼ 70.095δ + 1.729 + · · · , δ ≫ 1 ; λc ∼ 1.400 + 5.600 δ1/2 + 25.451 δ + · · · , δ ≪ 1 ; (Unit Slab) , (1.1.7a) λc ∼ 15.412δ + 1.001 + · · · , δ ≫ 1 ; λc ∼ 0.789 + 1.578 δ1/2 + 6.261 δ + · · · , δ ≪ 1 ; (Unit Disk) . (1.1.7b) Good agreement between (1.1.7) and full numerical results for λc computed from (1.1.4) is established. In particular, it is observed that the asymptotic result for λc in (1.1.7) derived for the δ ≫ 1 limit can give a reliable estimate of λc even for δ ≈ 0.03. The asymptotic results in (1.1.7) for δ ≪ 1 accurately predict λc for 0 < δ < 0.03 (see Fig. 3.5(b) and Fig. 3.6(b)). Therefore, for the unit slab and the unit disk, it is apparent that (1.1.7) gives a rather accurate estimate of λc for (1.1.4) for essentially the entire range 0 < δ < ∞, and hence (1.1.7) can give a good prediction of the pull-in voltage for (1.1.4). In §3.5.1 an analysis of the fold point location λc of (1.1.5) shows that the effect of a fringing-field is to reduce the pull-in voltage by an amount of O(δ) for δ ≪ 1. For the unit disk, it is calculated through a solvability condition that λc ∼ 0.789 − 0.160δ for δ ≪ 1. This asymptotic result is shown to compare very favorably with full numerical results computed from (1.1.5). In §3.5.2, a singular perturbation analysis of the fold point location λc for the an- nulus problem (1.1.6) in the limit δ → 0 shows that λc ∼ 0.789 + O (−1/ log δ). The coefficient of this logarithmic term, which is evaluated numerically, is shown to be pos- itive. Therefore, the effect of the inner disk of radius δ is to perturb the pull-in voltage for the membrane problem (1.1.2) rather significantly even when δ ≪ 1. Some related nonlinear eigenvalue problems with small holes were treated in [41] and [40]. Following on, Chapter 4 addresses the effects of the perturbations made in (1.1.4)- (1.1.6) to the upper branch of solutions of the unperturbed problem (1.1.2). It is 8 1.1. Micro-Electro Mechanical Systems observed by numerical calculation that in each case a positive value of δ acts to destroy the infinite fold point structure of the problem leaving behind a branch of solutions that undergoes a finite number of turning points before taking on limiting behaviour λ→ 0 as ||u||∞ → 1−. In the fringing fields case it was shown in [23] that for λ sufficiently small, (1.1.5) admits at least two solutions. This is in contrast to (1.1.2) which admits one unique solution, the minimal solution, for λ sufficiently small. In general, few rigorous results pertaining to the maximal solution branch of the fourth order problem (1.1.4) and the annulus problem (1.1.6) are available. A primary goal in the analysis of Chapter 4 is to develop a formal asymptotic anal- ysis, based on the method of matched asymptotic expansions, to provide an explicit analytical characterization of the asymptotic behaviour of the maximal solution branch to (1.1.4) in the limit ε = 1−||u||∞ → 0+, for which λ→ 0. This problem is studied for the unit slab, the unit disk and a certain class of general geometries Ω ⊂ R2. For these domains, explicit asymptotic expansions for λ as ε → 0 are derived for any δ > 0, and the results are shown to compare very favorably with full numerical results. In the case of the unit disc, the solution u to (1.1.4) in the limit ε→ 0 has a strong concentration near the origin owing to the nearly singular behaviour of the nonlinearity in (1.1.4). For more general two dimensional geometries, additional work is required to determine the location and multiplicity of concentration points. The singular perturbation analysis required to resolve these regions of concentration solution relies heavily on the system- atic use of logarithmic switchback terms. Such terms are notorious in the asymptotic analysis of PDE models arising in the study of low Reynolds number flows (cf. [27], [28], [36], [37]). The outline of this chapter is as follows. In § 4.1 a formal asymptotic approach is employed to obtain the asymptotic behaviour of the bifurcation curve to (1.1.3) in the unit ball in the limit ε ≡ 1 − ||u||∞ → 0+. In § 4.2 an explicit characterization of the maximal solution branch for the fringing-field problem (1.1.5) and for the biharmonic problem (1.1.4) in the limit ε ≡ 1− ||u||∞ → 0+ is developed for the unit slab. In § 4.3, similar results are given for (1.1.4) in the unit disk. In § 4.3.2 the limiting asymptotic behaviour of the maximal solution branches of (1.1.6) are constructed. In § 4.4, the problem of constructing the maximal solution branch of (1.1.4) for general Ω ⊂ R2 is discussed. A necessary condition for a solution to concentrate at some x0 ∈ Ω is proposed and the corresponding maximal branch is constructed in the limit as ε→ 0+ 9 1.2. Persistence on Patchy Domains - An Eigenvalue Optimization Problem 1.2 Persistence on Patchy Domains - An Eigenvalue Optimization Problem In the field of Ecology, a natural line of inquiry is that regarding the effect of habitat fragmentation on any occupying species. Fragmentation may occur naturally when weather changes ( i.e. droughts, floods ) facilitate the emergence of discontinuities in an organism’s preferred habitat. Although more frequently it is due to human activities such as agriculture, development and conservation to name a few. Mathematical models provide a natural framework to address such problems through reaction-diffusion theory. The diffusive logistic model, which describes the evolution of a population with density u(x, t) diffusing with constant diffusivity D > 0 throughout some habitat represented by a bounded domain Ω ⊂ R2, is formulated as ut = D∆u+ u [m(x)− u] , x ∈ Ω ; ∂nu = 0 , x ∈ ∂Ω ; (1.2.1a) u(x, 0) = u0(x) ≥ 0 , x ∈ Ω . (1.2.1b) The no-flux boundary condition in (1.2.1a) specifies that no individuals cross the bound- ary of the habitat Ω. The initial population density u0(x) is non-negative and not identically zero. The function m(x) represents the growth rate for the species, with m(x) > 0 in favourable parts of the habitat, and m(x) < 0 in unfavourable parts of the habitat. The integral ∫ Ωmdx measures the total resources available in the spatially heterogeneous environment. With respect to applications in ecology, this model was first formulated in [69]. To determine the stability of the extinction equilibrium solution u = 0, set u = φ(x)e−σt in (1.2.1), where φ(x)≪ 1, to obtain that φ satisfies ∆φ+ λm(x)φ = −σφ , x ∈ Ω; ∂nφ = 0 , x ∈ ∂Ω (1.2.2) where λ = 1/D > 0. The threshold for species persistence is determined by the stability border of the extinct solution u = 0. At this bifurcation point, the eigenvalue of the linearized problem about the zero solution must pass through zero. Therefore, by setting σ = 0 in (1.2.2) the problem reduces to the determination of a scalar λ and a function φ that satisfies the indefinite weight eigenvalue problem ∆φ+ λm(x)φ = 0 , x ∈ Ω ; ∂nφ = 0 , x ∈ ∂Ω ; ∫ Ω φ2 dx = 1 . (1.2.3) It is said that λ1 > 0 is a positive principal eigenvalue of (1.2.3) if the corresponding eigenfunction φ1 of (1.2.3) is positive in Ω. It is well-known (cf. [43], [55], [68]) that 10 1.2. Persistence on Patchy Domains - An Eigenvalue Optimization Problem (1.2.3) has a unique positive principal eigenvalue λ1 if and only if ∫ Ωmdx < 0 and the set Ω+ = {x ∈ Ω ; m(x) > 0} has positive measure. Such an eigenvalue is the smallest positive eigenvalue of (1.2.3). The positive principal eigenvalue λ1 is interpreted as the persistence threshold for the species. It is well-known that if λ < λ1, then u(x, t) → 0 uniformly in Ω̄ for all non-negative and non-trivial initial data, so that the population tends to extinction. Alternatively, if λ > λ1, then u(x, t) → u∗(x) uniformly in Ω̄ as t → ∞, where u∗ is the unique positive steady-state solution of (1.2.1). For this range of λ the species will persist. Many mathematical results for (1.2.1) under different boundary conditions are given in the pioneering works of [45], [46], and [47]. Related results for multi-species interactions and other mathematical problems in ecology are given in [48] (see also the survey article of [60]). An interesting problem in mathematical ecology is to determine, among all functions m(x) for which a persistence threshold exists, which m(x) yields the smallest λ1 for a fixed amount of total resources ∫ Ωmdx. In other words, we seek to determine the optimum arrangement of favourable habitats in Ω in order to allow the species to persist for the largest possible diffusivity D. This optimization problem was originally posed and studied in [45] and [47]. For (1.2.1) under Neumann boundary conditions in a two-dimensional domain Ω it was proved in [59] that the optimum m(x) is piecewise continuous and of bang-bang type. An earlier result showing the existence of a similar bang-bang optimal control for m(x) for the Dirichlet problem was given in [45]. For (1.2.1) posed in a one-dimensional interval 0 < x < 1 it was proved in [59] that the optimalm(x) consists of a single favourable habitat attached to one of the two endpoints of the interval. Related results were given in [47] under Dirichlet, Neumann, or Robin type boundary conditions. The minimization of λ1 in cylindrical domains of type (0, 1)×Ω ⊂ Rn for Ω ∈ Rn−1 was studied in [56]. It was shown that if | ∫Ωm(x) dx| is below some threshold value, the optimum λ1 occurs when the favourable habitat is concentrated near one of the corners of the domain. Otherwise, the optimum λ1 occurs when the favourable habitat is attached to either of the two ends of the domain with the shortest edge. For spatially periodic environments, the effect of fragmentation of the favourable resources was stud- ied in [42] using Steiner symmetrization, and some results were obtained for Dirichlet boundary conditions. Related applications of symmetrization ideas are given in [58]. A treatise on the modeling of biological invasions in periodic spatial environments is given [67]. In [64] stochastic methods were used to determine the persistence threshold for the diffusive logistic model for an infinitely periodic heterogeneous media. This study, which eliminated the effect of boundary conditions, showed that habitat fragmentation decreases the persistence of the species. For (1.2.1) in a bounded two-dimensional do- 11 1.2. Persistence on Patchy Domains - An Eigenvalue Optimization Problem main with Neumann boundary conditions, the existence of an optimal configuration for m(x) was proved in [63]. Moreover, numerical methods were used to show that the optimal favourable spatial habitats are either ball-shaped or strip-shaped, depending on the amount of available resources. Although these previous studies give considerable insight into the effect of spatial fragmentation of habitat resources on the persistence threshold in specific situations, such as cylindrical domains or periodic environments, the problem of the optimum choice for m(x) in arbitrary two-dimensional domains with no periodicity assumption is largely an open problem. In Chapter 5, the persistence threshold λ1 is calculated asymptotically, and then optimized, for a particular class of piecewise constant growth rate function m = mε(x) in an arbitrary two-dimensional domain. It is assumed that mε(x) is localized to n small circular patches of radii O(ε), each of which is centred either inside Ω or on ∂Ω. The boundary ∂Ω is assumed to be piecewise differentiable, but corners with nonzero angle are permitted on the domain boundary. The set of the centres of the interior patches is defined to be ΩI ≡ {x1, . . . , xn} ∩Ω while ΩB ≡ {x1, . . . , xn} ∩ ∂Ω is the set of the centres of the boundary patches. The patches are assumed to be well-separated in the sense that |xi − xj | ≫ O(ε) for i 6= j and that the interior patches are not too close to the boundary, i.e. dist(xj , ∂Ω) ≫ O(ε) whenever xj ∈ ΩI . To accommodate a boundary patch, an angle παj , representing the angular fraction of a circular patch that is contained within Ω, is associated with each xj for j = 1, . . . , n. More specifically, αj = 2 whenever xj = Ω I , αj = 1 when xj ∈ ΩB and xj is a point where ∂Ω is smooth, and αj = 1/2 when xj ∈ ∂Ω is at a π/2 corner of ∂Ω, etc. The growth rate function m = mε(x) in (1.2.3) is taken to have the specific form m = mε(x) ≡   mj/ε 2 , x ∈ Ωεj , j = 1, . . . , n , −mb , x ∈ Ω\ ⋃n j=1Ωεj . (1.2.4) Here Ωεj ≡ {x | |x− xj | ≤ ερj ∩ Ω}, so that each patch Ωεj is the portion of a circular disk of radius ερj that is strictly inside Ω. The constant mj is the local growth rate of the jth patch, with mj > 0 for a favourable habitat and mj < 0 for a non-favourable habitat. The constant mb > 0 is the background bulk decay rate for the unfavourable habitat. In terms of this growth rate function, the condition of [43], [55], and [68] for the existence of a persistence threshold is that one of the mj for j = 1, . . . , n must be positive, and that the asymptotically valid inequality ∫ Ω mε dx = −mb|Ω|+ π 2 n∑ j=1 αjmjρ 2 j +O(ε2) < 0 . (1.2.5) 12 1.2. Persistence on Patchy Domains - An Eigenvalue Optimization Problem holds for the total resources. Here |Ω| denotes the area of Ω. We assume that the parameters are chosen so that (1.2.5) is satisfied. A schematic plot of a domain with circular patches is shown in Fig. 1.4 Ω − + + − Figure 1.4: Schematic plot of a two-dimensional domain with localized strongly favourable (+) or unfavourable (−) habitats or patches, as described by (1.2.4). The patches inside the domain are small circular disks. On the domain boundary, the patches are the portions of circular disks that lie within the domain. This specific form for mε(x) is motivated by Theorem 1.1 of [59] which states that the optimal growth rate function must be of bang-bang type, and the result of [63] which shows that a sufficiently small optimum favourable habitat must be a circular disk. In § 5.1 the method of matched asymptotic expansions is used to derive a two-term asymptotic expansion for the persistence threshold λ1 for the case of either a single favourable interior or boundary habitat. The asymptotic analysis is extended in § 5.2 to calculate asymptotically λ1 for (1.2.3) with growth rate function (1.2.4), which allows for multiple interior or boundary habitats. The analysis, which is summarized at the end of §5.2, shows that λ1 has the two-term asymptotic expansion λ1 = µ0ν + ν 2µ1(x1, . . . , xn) +O(ν3) , ν(ε) = −1/ log ε . (1.2.6) Here the leading-order coefficient µ0 is the unique positive root of B(µ0) = 0 on 0 < µ0 < 2/(mJρ 2 J), where B(µ0) ≡ −mb|Ω|+ π n∑ j=1 αjmjρ 2 j 2−mjρ2jµ0 , mJρ 2 J ≡ max mj>0 {mjρ2j | j = 1, . . . , n } . (1.2.7) The coefficient µ1, which depends explicitly on the spatial configuration {x1, . . . , xn} 13 1.2. Persistence on Patchy Domains - An Eigenvalue Optimization Problem of patches, is determined in terms of a matrix involving the Neumann Green’s function and the surface Neumann Green’s function for Ω. In § 5.3 the effects of resource fragmentation on the coefficients µ0 and µ1 of the asymptotic expansion of the persistence threshold are analyzed. For a prescribed amount of resources, for which ∫ Ωmε dx in (1.2.5) is fixed, the patch configuration that minimizes µ0, or in certain degenerate situations, minimizes the coefficient µ1 in (1.2.6), is determined. In §5.3.2 an analysis of the leading-order coefficient µ0 in (1.2.6) is presented. This coefficient provides a surprisingly large amount of information on the relationship be- tween habitat fragmentation and the persistence threshold. There are several key qual- itative principles that are established. First, the fragmentation of a favourable interior habitat into two smaller favourable interior habitats is shown to be deleterious to species persistence, whereas the migration of an interior favourable habitat to the boundary of the domain is always advantageous. The optimal boundary location to concentrate a favourable resource is at a corner of the domain boundary with the smallest opening an- gle. Second, the fragmentation of a favourable interior habitat into a smaller favourable interior habitat together with a favourable boundary habitat is advantageous to species persistence when the boundary habitat is sufficiently strong. Further general principles, based on the optimization of µ0, are summarized in Qualitative Results I–III of § 5.3.2. In addition, an illustration of these principles for certain patch distributions in the unit disk is given. In § 5.3.3 it is also demonstrated that in certain degenerate situations, the problem of determining the optimal location for a favourable resource requires the examination of the coefficient µ1 of the second term in the asymptotic expansion of λ. In particular, such a problem occurs in optimizing λ with respect to the boundary location of a single favourable boundary patch in a domain with a smooth boundary. In this case, it is shown that λ is minimized when the boundary patch is centred at a point x0 ∈ ∂Ω at which the regular part of the surface Neumann Green’s function attains its global maximum value on the boundary. The relationship between the global maximum of the boundary curvature and the regular part of the surface Neumann Green’s function for smooth perturbations of the unit disk is investigated. In § 5.3.3 the optimization of λ for the case where an additional favourable resource is to be located inside a domain with a pre-existing and fixed patch distribution is considered. In this case, the optimization of λ typically requires the examination of the coefficient µ1 of the second-order term in the asymptotic expansion of λ. The theory is illustrated for two specific examples involving the unit disk and the unit square, for which the required Green’s functions are known analytically. Related problems involving the asymptotic calculation and optimization of the fun- 14 1.2. Persistence on Patchy Domains - An Eigenvalue Optimization Problem damental eigenvalue of the Laplacian have been studied in perforated two-dimensional domains (cf. [51], [57], [61], [41], and [71]), in two-dimensional domains with perforated boundaries (cf. [44], [52], [53], [62]), and under the effect of strongly localized potentials (cf. [54]). 15 Chapter 2 Mathematical Modeling of Micro-Electro Mechanical Systems In this section, models of the MEMS capacitor device introduced in §1.1 will be pre- sented. In §2.1 the original analysis from [4] is presented. In § 2.2 a more contemporary model, introduced by Pelesko [33], will be detailed along with some preliminary results. 2.1 Lumped Mass Spring Model In their initial work on MEMS, Nathanson et al. [4] concluded that allowing the de- flection of the beam to be a function of space would complicate the resulting analysis unnecessarily and obscure the simplicity of the device. For this reason, they assumed that beam deflection was a function of time only and formulated a model in which the deflecting plate was represented by a mass on a spring whose spring constant, k, would approximately characterize the mechanical properties of the beam. The imagined configuration is represented schematically in Figure 2.1. d u Elastic plate of mass m at potential V . Ground plate of area A. Figure 2.1: Schematic diagram of lumped mass spring model of MEMS deflection. The upper plate is assumed to be of mass m, surface area A and at potential V . To determine a differential equation satisfied by u(t), Newton’s Second Law of mo- tion is applied to all forces acting on the plate. There are three forces to be considered; the restoring force of the Hookean spring, Fs = −k(u − l) where l is the rest length of 16 2.1. Lumped Mass Spring Model the spring, some damping effect assumed to be proportional to the velocity of the plate, that is Fd = −aut and an electrostatic force Fe between the two plates. To calculate Fe, note that the potential φ between the two plates satisfies ∆φ = 0, 0 < z′ < d− u; φ(d− u) = V φ(0) = 0 (2.1.1) which generates an electric field E = −∇φ with resulting total energy U = ǫ0 2 ∫ S |E|2 where S is the volume of the capacitor and ǫ0 is the permittivity of free space. By op- erating in the small aspect ratio regime d/L≪ 1, equation (2.1.1) is well approximated by φz′z′ = 0 and is therefore easily solved to give the linear potential φ = V z′ d− u between the plates. The electric field E = −φz′ is then constant - the standard approx- imation for a parallel plate capacitor based on a small aspect ratio. The expression for the total energy is now reduced to U = ǫ0 2 ∫ S V 2 (d− u)2 = ǫ0AV 2 2(d − u) and differentiation with respect to (d−u) reveals that the electrostatic force is given by Fe = ǫ0 2 AV 2 (d− u)2 . Combining all the forces in Newton’s Equation of motion, the ODE m d2u dt2 + a du dt + k(u− l) = ǫ0 2 AV 2 (d− u)2 . (2.1.2) is obtained for the displacement u(t) of the upper plate. This equation is now reduced in complexity by introducing non-dimensional variables. The time scale of the system is chosen to reflect that viscosity is the dominant resistive force in the system. In other words, the primary restorative force acting against the electrostatic force is damping. This motivates the introduction of variables t′ = k a t, u′ = u− l d− l (2.1.3) 17 2.1. Lumped Mass Spring Model which reduces (2.1.2) to α2 d2u dt2 + du dt + u = λ (1− u)2 . (2.1.4) after dropping the primes where α = √ mk a , λ = 1 2 ǫ0AV 2 k(d− l)3 . (2.1.5) The parameter α is known in engineering parlance as the quality factor or Q for the system. The quality factor compares the rate at which a system oscillates to the rate at which it dissipates energy. In a viscosity dominated system, a small value of Q is to expected corresponding to a system in which oscillatory behavior is quickly damped out. The parameter λ, proportional to V 2, is a ratio of electrostatic forces to spring forces and acts as a natural bifurcation parameter for the system. By setting time derivatives to zero in equation (2.1.2), the equilibrium spacing of the plates is shown to satisfy λ = u(1 − u)2 where physical constraints fix 0 ≤ u < 1. The steady state defections are readily determined from this relationship and conveniently displayed in the bifurcation diagram, Fig. 2.2. 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.03 0.06 0.09 0.12 0.15 u λ (λ∗, u∗) Figure 2.2: Bifurcation diagram of steady state solutions to equation (2.1.2). A saddle-node or fold bifurcation is observed at the point (λ∗, u∗) = (4/27, 1/3) with a stable (solid) and an unstable (dashed) solution branch emanating from the bifurcation point. From the bifurcation diagram, it is observed that for a certain range of λ, there exists one stable and one unstable branch ( linear stability is easily determined ). It is also observed that when λ is increased beyond some critical value λ∗, a steady state deflection is no longer achieved and therefore the top plate will touchdown on the ground plate. The location of this saddle-node or fold bifurcation is determined by the condition du/dλ = 0 and found to be λ∗ = 4/27 with a corresponding u∗ = 1/3. In 18 2.2. A Model From Linear Elasticity terms of dimensional quantities, this bifurcation point is given by u∗ = 1 3 (d− l), V ∗ = √ 8 27 k(d− l)3 ǫ0A . The assumption that the deflection of the plate is a function of time alone allows a tractable analysis of the steady state problem, although, it cannot take into account the geometry or the elastic properties of the deflecting plate. Despite these shortcomings, the model is still useful as it predicts qualitative properties of the device including touchdown for voltages above the critical V ∗; a quantity for which a compact expression is provided. 2.2 A Model From Linear Elasticity To better accommodate the geometry and elastic nature of the deflecting plate, Pelesko [33] developed a model which coupled equations of linear elasticity with those of elec- trostatics. In addition, a spatially varying dielectric permittivity profile ǫ1(x, y) in the deflecting plate is accounted for, which in turn requires that a potential φ1 be deter- mined in the gap and another potential φ2 determined inside the plate. In dimensional form, the model requires that the deflection u(x, y, t) of a plate occupying a region Ω ⊂ R2 with smooth boundary ∂Ω, satisfies ρh ∂2u ∂t2 + a ∂u ∂t +D∆2⊥u− T ∆⊥u = − ǫ1 2 |∇φ2|2, z = u(x, y, t) − h/2 (2.2.1) ∆φ1 = 0, 0 < z < u(x, y, t) − h/2 (2.2.2) ∇ · (ǫ1∇φ2) = 0 |z − u(x, y, t)| < h/2, (2.2.3) The potential at the ground plate is zero and a voltage V is applied on the top of the upper plate so that φ1(0) = 0, (2.2.4) φ2(u+ h/2) = V. (2.2.5) The specification of the potential is completed by demanding continuous differentiability at the boundary of the gap and upper plate. Therefore φ1 = φ2, ǫ1∇φ2 · n̂ = ǫ0∇φ1 · n̂ z = u(x, y, t)− h/2 (2.2.6) 19 2.2. A Model From Linear Elasticity where n̂ is the normal vector to the surface z = u(x, y, t) − h/2. In equation (2.2.1), D is the flexural rigidity, ρ and h are the density and thickness of the deflecting plate respectively while T represents the tension on the plate. The value of a reflects the strength of damping in the system while ǫ0 is the permittivity of free space. The subscript ⊥ indicates differentiation restricted to the xy plane. Specifically, we have ∆ ≡ ∂ 2 ∂x2 + ∂2 ∂y2 + ∂2 ∂z2 , ∆⊥ ≡ ∂ 2 ∂x2 + ∂2 ∂y2 , ∇ ≡ ( ∂ ∂x , ∂ ∂y , ∂ ∂z ) . There are several options for the conditions satisfied by u(x, y, t) on the boundary ∂Ω, though a common requirement is that the plate is clamped which requires u = d, ∂nu = 0, (x, y) ∈ ∂Ω (2.2.7) where ∂nu is the outward facing normal derivative with respect to ∂Ω. The problem may be decoupled by taking advantage of the small aspect ratio σ ≡ d/L. Focusing on the equations for the potential for now, non-dimensionalize with x′ = x L y′ = y L z′ = z d , u′ = u d , φ′1 = φ1 V , φ′2 = φ2 V , (2.2.8) gives after dropping primes, ∂2φ1 ∂z2 + σ2 ( ∂2φ1 ∂x2 + ∂2φ1 ∂y2 ) = 0, 0 < z < u− h 2d (2.2.9) ǫ1 ∂2φ2 ∂z2 + σ2 ( ∂ ∂x ( ǫ1 ∂φ2 ∂x ) + ∂ ∂y ( ǫ1 ∂φ2 ∂y )) = 0, |z − u| < h 2d (2.2.10) with continuous differentiability enforced by φ1 = φ2, ǫ1 ∂φ2 ∂z − ǫ0 ∂φ1 ∂z = σ2 ( ǫ1 ( ∂u ∂x ∂φ2 ∂x + ∂u ∂y ∂φ2 ∂y ) − ǫ0 ( ∂u ∂x ∂φ1 ∂x + ∂u ∂y ∂φ1 ∂y )) (2.2.11) when z = u− h/2d. Solving this system to leading order ( i.e. σ ≪ 1 ) reveals that the potential is well approximated by a linear function of z in each region. Applying the relevant boundary and continuity conditions this potential is found to be φ =   ǫ1z ǫ1(u− h/2d) + ǫ0(h/d) 0 < z < u− h/2d 1 + ǫ0(z − (u+ h/2d)) ǫ1(u− h/2d) + ǫ0h/d u− h/2d < z < u+ h/2d (2.2.12) With this explicit representation of φ, the right hand side of equation (2.2.1) satisfies 20 2.2. A Model From Linear Elasticity |∇φ2| = (∂zφ2)2 +O(σ2) on z = u− h/2d where ∂zφ2 = ǫ0 ǫ1(u− h/2d) + ǫ0h/d By applying the thin plate limit, h/d→ 0, this expression simplifies to ∂zφ2 = ǫ0 ǫ1u Finally, a time scale dominated by viscous effects, t = k/a, is chosen and so (2.2.1) can be written in non dimensional form as α2 ∂2u ∂t2 + ∂u ∂t + δ∆2⊥u−∆⊥u = − λf(x, y) u2 (2.2.13) with boundary conditions u = 1, ∂nu = 0, (x, y) ∈ ∂Ω (2.2.14) The dimensionless quantities α, δ, λ and f(x, y) are defined as follows: α = √ Tρh aL , δ = D TL2 , λ = ǫ0L 2V 2 2d3T , f(x, y) = ǫ0 ǫ1(x, y) (2.2.15) where α is the quality factor for the system, δ measures the relative importance of tension and rigidity, λ is a ratio of electrostatic and elastic forces in the system while f(x, y) is the permittivity of the plate relative to that of free space. The parameter λ is proportional to the square of the applied voltage, V 2, and because the operation of a MEMS device may necessitate V varying over some range, λ is a natural bifurcation parameter for the system. In addition, it should be noted that λ ≥ 0 for all physical situations. For the convenience of homogeneous boundary conditions, the transformation u 7→ 1+u is made and the ⊥ notation is dropped as all differentiation is now restricted to the xy plane. The three dimensional problem has now been reduced to a two dimensional problem where z = u(x, y) and α2 ∂2u ∂t2 + ∂u ∂t + δ∆2u−∆u = −λf(x, y) (1 + u)2 (x, y) ∈ Ω; u = ∂nu = 0 (x, y) ∈ ∂Ω (2.2.16) The focus of our attention is further restricted to the case of small quality factor for which the α2utt term in (2.2.16) is considered negligible. This approximation, called the viscous damping limit, assumes that inertial effects are negligible compared to those of damping. By providing some initial data u0(x, y) at time t = 0 the deflection z = 21 2.2. A Model From Linear Elasticity u(x, y, t) of the upper plate occupying a region Ω ⊂ R2 with smooth boundary ∂Ω satisfies ∂u ∂t = −δ∆2u+∆u− λf(x, y) (1 + u)2 , (x, y) ∈ Ω ; u = ∂nu = 0, (x, y) ∈ ∂Ω u(x, y, 0) = u0(x, y), (x, y) ∈ Ω (2.2.17) Equation (2.2.17) represents a model for electrostatically actuated MEMS devices in its most general form. In subsequent studies of this equation, further simplifications have been employed to reduce the complexity and permit more ready analysis. In the original work of Pelesko ([33]), the approximations δ = 0 and f(x, y) = 1 were adopted, thus approximating the deflecting plate as a membrane with a homogeneous dielectric profile. Omitting the boundary condition ∂nu = 0 on ∂Ω, the reduced equation is ∂u ∂t = ∆u− λ (1 + u)2 , (x, y) ∈ Ω ; u = 0, (x, y) ∈ ∂Ω; u(x, y, 0) = u0(x, y). (2.2.18) Steady state and dynamical solutions to equation (2.2.18) have received significant at- tention recently and have been shown to exhibit a rich collection of mathematical phe- nomena [33],[13],[9]. In §3.1, some preliminary results on the steady state equation ∆u = λ (1 + u)2 , (x, y) ∈ Ω ; u = 0, (x, y) ∈ ∂Ω (2.2.19) are presented. Another interesting variation of the basic membrane equation (2.2.18) is the so-called fringing fields problem, introduced in [38]. In this problem, the radially symmetric deflection of a membrane with homogeneous dielectric permittivity profile ( f(x, y) = 1 ) is considered in a drum shaped device (r, z) ∈ [0, L] × [0, d]. The formulation of this problem requires careful determination of the electric potential in the gap up to and including the O(σ2) term. The equation for the potential in the gap, in dimensionless variables, is ∂2φ ∂z2 + σ2 ( ∂2φ ∂r2 + 1 r ∂φ ∂r ) = 0, (r, z) ∈ [0, 1] × [0, 1]; φ(1, z) = 0 φ(r, 1) = 1, φ(r, 0) = 0 (2.2.20) To satisfy the condition at r = 1, a local analysis in the vicinity of the boundary is required. After a little work, the following uniformly valid asymptotically approximation 22 2.2. A Model From Linear Elasticity of φ is obtained φ(r, z) = z u(r) + 2 π ∞∑ n=1 (−1)n n e−npi( 1−r σ ) sin(nπz) (2.2.21) as so |∇φ2∣∣ z=u = ( ∂φ ∂z )2 ∣∣∣ z=u + σ2 ( ∂φ ∂r )2 ∣∣∣ z=u = 1 + σ2 u′2 u2 + · · · This leads to the following model for the equilibrium deflection of a membrane MEMS device ∆u = λ(1 + σ2|∇u|2) (1 + u)2 , (x, y) ∈ Ω u = 0 (x, y) ∈ ∂Ω where, as usual, σ = d2/L2. 23 Chapter 3 Fold Point Asymptotics This chapter breaks down as follows. In § 3.1 the methods employed to obtain numerical solutions to the various nonlinear eigenvalue problems describing MEMS, are outlined. Some of the key qualitative features of the numerical solutions are discussed. In partic- ular, it is observed that in each of the perturbed equations; the beam (1.1.4), fringing fields (1.1.5) and the annulus (1.1.6), the location of the primary fold point, located at the end of the minimal branch, exhibits significant deviation from the unperturbed problem. In § 3.2, an asymptotic expansion for the location of the perturbed fold point is formulated for (1.1.4) and the case Ω = (−1, 1). In the limit δ → 0, a boundary layer of width δ1/2 is required to enforce the derivative boundary condition which in turn provides boundary conditions to problems in the r = O(1) region. In § 3.3, the analysis is extended to develop an asymptotic formulation of the principal fold point for (1.1.4) on an arbitrary 2D domain. In § 3.4 the limit δ → ∞ is considered for (1.1.4) and an expansion of the fold point is developed. This case is somewhat simpler that the δ → 0 limit as only a regular expansion is required. When Ω reduces to the unit disc or unit slab, asymptotic results are verified with full numerics and interestingly, by combining the results for δ → 0 and δ → ∞, the asymptotic predictions of the fold point are observed to be accurate for the entire range 0 < δ <∞. In § 3.5, similar asymptotic expansions for the principal fold point are established for the fringing field (1.1.5) and the annulus problem (1.1.6). In the annulus case, it is observed that in the limit δ → 0, the fold point is deflected by a quantity O(log−1 δ), indicating that a change in domain topology has a significant effect on the onset of the pull-in instability. 3.1 Numerical Solution Nonlinear Eigenvalue Problems In this section, an outline of the numerical methods used to compute the bifurcation diagrams associated with the nonlinear eigenvalue problems (1.1.4), (1.1.2), (1.1.5), and (1.1.6), is provided. The results of these computations provide motivation for the asymptotic analysis in §3.2-3.5 and are useful for validating our asymptotic results. For the membrane problem (1.1.2), the use of scale invariance as a computational 24 3.1. Numerical Solution Nonlinear Eigenvalue Problems technique to compute bifurcation diagrams was explored in [34]. By introducing the new variables t and w by u(r) = −1 + αw(y) , y = ηr , it was shown in [34] that the bifurcation diagram for (1.1.2) can be parameterized as |u(0)| = 1− 1 w(η) , λ = η2 w3(η) , η > 0 , (3.1.1) where w(η) is the solution of the initial value problem w′′ + 1 y w′ = 1 w2 , 0 < y < η ; w(0) = 1 , w′(0) = 0 . (3.1.2) By solving (3.1.2) numerically, the bifurcation diagram, as shown in Fig. 3.1, is obtained. For this problem, it was shown in [34] that there are an infinite number of fold points that have the limiting behaviour |u(0)| → 1− as λ → 4/9. A similar scale invariance method can also be used for computing solutions to the generalized membrane problem (1.1.3). 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 |u(0)| λ (a) Unit Disk: Membrane Problem 0.4400 0.4425 0.4450 0.4475 0.4500 0.975 0.980 0.985 0.990 0.995 1.000 |u(0)| λ (b) Unit Disk (Zoomed): Membrane Problem Figure 3.1: Numerical solutions for |u(0)| versus λ computed from (1.1.2) for the unit disk Ω = {|x| ≤ 1} in two-dimensions. The magnified figure on the right shows the beginning of the infinite set of fold points. Next, the scale invariance method is extended to compute solutions to the fourth- order problem (1.1.4) for the two-dimensional unit disk. By introducing new variables v and y by u = −1 + αv(y) , y = Tr , 25 3.1. Numerical Solution Nonlinear Eigenvalue Problems equation (1.1.4) becomes −δαT 4∆2yv + αT 2∆yv = λ α2v2 . The conditions u′(0) = u′′′(0) = 0 imply that v′(0) = v′′′(0) = 0. The free parameter v(0) is chosen as v(0) = 1 so that u(0) = −1 + α. Enforcing the boundary condition u(1) = 0 requires that α = 1/v(T ), while u′(1) = 0 is satisfied if v′(T ) = 0. Finally, by letting λ = α3T 4, a parametric form of the bifurcation diagram is given by |u(0)| = 1− 1 v(T ) , λ = T 4 v3(T ) , where v(y) is the solution of −δ∆2yv + 1 T 2 ∆yv = 1 v2 , 0 < y < T ; v(0) = 1 , v′(0) = 0 , v′′′(0) = 0 , v′(T ) = 0 . (3.1.3) There are two options for solving (3.1.3). The first option, representing a shooting approach, consists of solving (3.1.3) as an initial value problem and choosing the value of v′′(0) so that v′(T ) = 0. The second option is to solve (3.1.3) directly as a boundary value problem. For this approach it is convenient to rescale the interval to [0, 1] by making the transformation y → Ty, resulting in −δ∆2yv +∆yv = T 4 v2 , 0 ≤ y ≤ 1 ; v(0) = 1 , v′(0) = 0 , v′′′(0) = 0 , v′(1) = 0 . (3.1.4) In these variables, the bifurcation diagram for (1.1.4) is then parameterized in terms of T by |u(0)| = 1− 1 v(1) , λ = T 4 v3(1) , (3.1.5) where v(y) is the solution to (3.1.4). A similar approach is used to compute the bifur- cation diagram of (1.1.4) in a slab. It is remarked that the solution of the membrane problem (1.1.2) using the scale invariance method requires that the initial value problem (3.1.2) be solved exactly once. In contrast, for the fourth-order problem (1.1.4) the solution of either (3.1.3) or (3.1.4) must be computed for each point on the bifurcation branch. However, notice that in contrast to solving (1.1.4) directly using a two-parameter shooting method, the scale invariance method leading to (3.1.3) involves only a one-parameter shooting. In Fig. 3.2 the numerically computed bifurcation diagram of |u(0)| versus λ is plot- 26 3.1. Numerical Solution Nonlinear Eigenvalue Problems λ |u(0)| 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 (a) Unit Disk λ |u(0)| δ = 0.0001 δ = 0.01 δ = 0.05 δ = 0.1 0.90 0.92 0.94 0.96 0.98 1.00 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (b) Unit Disk (Zoomed) Figure 3.2: Numerical solutions of (1.1.4) for the unit disk Ω = {|x| ≤ 1} for several values of δ. From left to right the solution branches correspond to δ = 0.0001, 0.01, 0.05, 0.1. The figure on the right is a magnification of a portion of the left figure. ted for Ω = {|x| ≤ 1} and for various small positive values of δ. These numerical results indicate that the presence of the biharmonic term in (1.1.4) with small nonzero coeffi- cient δ destroys the infinite fold point behaviour associated with the membrane problem (1.1.2) shown previously in Fig. 3.1. Furthermore, numerical results suggest the exis- tence of some critical value δc ≪ 1, such that for δ > δc equation (1.1.4) exhibits either zero, one, or two, solutions, with the resulting bifurcation diagram having only one fold point at the end of the minimal solution branch. In §3.3 an asymptotic expansion of the fold point at the end of the minimal solution branch for (1.1.4) in powers of δ1/2 for δ ≪ 1 is developed. In §3.2 the corresponding problem in the unit slab is considered. In §3.4 an asymptotic expansion of the fold point for (1.1.4) when δ ≫ 1 for both the unit disk and the unit slab is constructed. To numerically compute the bifurcation diagram associated with the fringing-field problem (1.1.5) in the unit disk the numerically observed fact that the solution can be uniquely characterized by the value of u(0) is exploited. By assigning a range of values to u(0) in the interval (−1, 0), (1.1.5) is solved as an initial value problem and the value of λ is uniquely determined by the zero of g(λ) = u(1;λ). A Newton iteration scheme is implemented on g(λ) with initial guess u(0) = λ = 0. This method was found to be effective provided the stepsize in u(0) is sufficiently small. In order to numerically treat the annulus problem (1.1.6) it is advantageous to rescale the domain to [0, 1] with the change of variables ρ = (r − δ)/(1 − δ). Then (1.1.6) transforms to d2u dρ2 + (1− δ) δ + (1− δ)ρ du dρ = λ(1− δ)2 (1 + u)2 , 0 < ρ < 1 ; u(0) = u(1) = 0 . (3.1.6) In a way similar to the numerical approach for the fringing-fields problem (1.1.5), so- lutions to (3.1.6) are computed at predetermined values of u′(0) < 0 so that (3.1.6) 27 3.1. Numerical Solution Nonlinear Eigenvalue Problems becomes an initial value problem. The value of λ is then fixed by the unique zero of g(λ) = u(1;λ), which is computed using Newton’s method. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 λ |u(0)| (a) Unit Disk: Fringing Field 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 1.0 λ ||u||∞ (b) Annulus Figure 3.3: Left figure: Numerical solutions for |u(0)| versus λ computed from the fringing-field problem (1.1.5) for the unit disk Ω = {|x| ≤ 1} with, from left to right, δ = 1, 0.5, 0.1. Right figure: Numerical solutions for ||u||∞ versus λ computed from (1.1.6) in the annulus δ < r < 1 with, from left to right, δ = 0.00001, 0.001, 0.1. The numerically computed bifurcation diagrams for the fringing-fields problem (1.1.5) and the annulus problem (1.1.6) are plotted in Fig. 3.3(a) and Fig. 3.3(b), respectively, for various values of δ. It is again observed that, for δ small, that the effect of the per- turbation is to destroy the infinite fold point behaviour associated with the membrane problem (1.1.2). In §3.5 asymptotic results for the location of the fold point at the end of the minimal solution branch are given for (1.1.5) and (1.1.6) when δ ≪ 1. A straightforward approach to compute solutions to (1.1.4), (1.1.5), and (1.1.6), is to solve the underlying ODE boundary value problems by using a standard boundary value solver such as COLSYS [1]. This approach works well provided that the bifurcation diagram can be parameterized in terms of the coordinate on the vertical axis of the bifurcation diagram, such as u(0). Then, the BVP solver can be formulated to solve for u(x) and λ. Next, a more general approach to the numerical solution of the bifurcation branch of (1.1.4) is described, which also applies to a multi-dimensional domain Ω. For the unit square Ω = [0, 1] × [0, 1], (1.1.4) is not amenable to the scale invariance technique and a more general approach based on pseudo-arclength continuation (cf. [24]) is required. This method takes a direct approach to compute solutions of the general problem f : Rn × R → Rn, f(u, λ) = 0 . (3.1.7) Starting with an initial solution (u0, λ0), the method seeks to determine a sequence of 28 3.1. Numerical Solution Nonlinear Eigenvalue Problems points (uj , λj) which satisfy (3.1.7) to within some specified tolerance. The following is a brief outline of this method based on [24]. In order to compute solutions around fold points at which the system has a singular jacobian and the bi- furcation curve has a vertical tangent, the method introduces a parameterization of the curve n(u(s), λ(s), ds) = 0 in terms of an arclength parameter s and seeks new points on the solution branch at predetermined values of the steplength ds. To choose n(u(s), λ(s), ds) = 0, consider some accepted point (uj , λj) and its tangent vector to the curve at that point (u̇j , λ̇j), where an overdot represents differentiation with respect to arclength s. Now, define n(u(s), λ(s), ds) = u̇Tj · (u− uj) + λ̇j(λ− λj)− ds , (3.1.8) as the hyperplane whose normal vector is (u̇j, λ̇j) and whose perpendicular distance from (uj , λj) is ds. The intersection of this hyperplane with the bifurcation curve will be non-zero provided the curvature of the branch and ds are not too large. With this specification of n, the pseudo-arclength continuation method seeks a solution to the augmented system f(u(s), λ(s)) = 0 , n(u(s), λ(s), s) = 0 , (3.1.9) which is non-singular at simple fold points (cf. [24]). Applying Newton’s method with initial guess (uj , λj) to the solution of (3.1.9) results in the following iteration scheme:  fu(u(k), λ(k)) fλ(u(k), λ(k)) u̇Tj λ̇j     ∆u ∆λ   = −   f(u(k), λ(k)) n(u(k), λ(k))   u(k+1) = u(k) +∆u λ(k+1) = λ(k) +∆λ . (3.1.10) By differentiating (3.1.7) with respect to λ and solving the resulting linear system fuuλ + fλ = 0, the tangent vector (u̇, λ̇) is specified as u̇ = auλ, λ̇ = a, a = ±1√ 1 + ||uλ||22 . The sign of a is chosen to preserve the direction in which the branch is traversed. To compute solutions of (1.1.4) by the pseudo-arclength method in the unit square Ω = [0, 1] × [0, 1], the partial derivatives are approximated by central finite difference quotients, which results in a large system of nonlinear equations. In Fig. 3.4(b) the numerically computed bifurcation diagram for (1.1.4) in the unit square is plotted for 29 3.1. Numerical Solution Nonlinear Eigenvalue Problems several values of δ. The computations were done with a uniform mesh-spacing of h = 1/100 in the x and y directions. The bifurcation diagram is similar to that of the unit disk shown in Fig. 3.2(a). In Fig. 3.4(a) the corresponding numerical bifurcation diagram for (1.1.4) in the one-dimensional unit slab is plotted. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 5.0 10.0 15.0 20.0 25.0 λ |u(0)| (a) Unit Slab 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.0 0.2 0.4 0.6 0.8 1.0 λ |u(0)| (b) Unit Square Figure 3.4: Numerical solutions of (1.1.4) for the slab 0 < x < 1 (left figure) and the unit square for several values of δ. From left to right the solution branches correspond to δ = 0.1, 1.0, 2.5, 5.0 (left figure) and δ = 0.0001, 0.001, 0.01 (right figure). A quantitative asymptotic theory describing the destruction of the infinite fold points for (1.1.2) when (1.1.2) is perturbed for 0 < δ ≪ 1 to either the biharmonic problem (1.1.4), the fringing-field problem (1.1.5), or the annulus problems (1.1.6), is given in Ch. 4. This is done by constructing the limiting form of the bifurcation diagram when ||u||∞ = 1− ε, where ε→ 0+. Asymptotic results for the limiting behaviour λ→ 0 and ε→ 0 of the maximal solution branch are also presented for these perturbed problems. 3.1.1 Simple Upper Bounds For λc In the case where Ω represents either the unit slab or the unit disk, a simple upper bound for the fold point location at the end of the minimal solution branch, λc, is obtained for (1.1.4). The existence of this bound demonstrates that λc is finite and provides a rather good estimate of its value. The bound is established in terms of the principal eigenvalue of the differential operator appearing on the left hand side of (1.1.4). Therefore the associated eigenvalue problem requires the determination of a function φ and a scalar µ such that − δ∆2φ+∆φ = −µφ , x ∈ Ω ; φ = ∂nφ = 0 , x ∈ ∂Ω . (3.1.11) When Ω is either the unit slab or the unit disk the positivity of the first eigenfunction φ0 is verified numerically from the explicit formulae for φ0 given below in (3.1.16) and (3.1.17). Owing to the lack of a maximum principle, the positivity of the first 30 3.1. Numerical Solution Nonlinear Eigenvalue Problems eigenfunction for (3.1.11) is not guaranteed for more general domains. In particular, for domains such as squares or rectangles or annuli, the principle eigenfunction of the limiting problem δ → ∞ in (3.1.11) is known to change sign (cf. [6], [7]). For a survey of such results see [39]. Therefore, the following discussion is limited to either the unit slab or the unit disk. To derive an upper bound for λc, the approach in [34] needs to be modified only slightly. We assume that u exists and use Green’s second identity on u and the principal eigenfunction φ0 and eigenvalue µ0 of (3.1.11) to obtain 0 = δ ∫ Ω (−φ0∆2u+ u∆2φ0) dx = ∫ Ω φ0 ( λ (1 + u)2 + µ0u ) dx− ∫ Ω (φ0∆u− u∆φ0) dx . (3.1.12) The second integral on the right-hand side of (3.1.12) vanishes identically, and so a necessary condition for a solution to (1.1.4) is that ∫ Ω φ0 ( λ (1 + u)2 + µ0u ) dx = 0 . (3.1.13) Since φ0 > 0, then there is no solution to (3.1.13) when λ (1 + u)2 + µ0u > 0, ∀u > −1 . (3.1.14) By considering the point at which the inequality (3.1.14) ceases to hold, it is clear that there is no solution to (1.1.4) when λ > 4µ0/27 and therefore λc ≤ λ̄ ≡ 4µ0 27 , (3.1.15) where µ0 is the first eigenvalue of (3.1.11). For the unit slab 0 < x < 1, a simple calculation shows that the eigenfunctions of (3.1.11) are given up to a scalar multiple by φ = cosh(ξ1x)− cos(ξ2x)− ( ξ2 sinh(ξ1x)− ξ1 sin(ξ2x) ξ2 sinh ξ1 − ξ1 sin ξ2 ) (cosh ξ1 − cos ξ2) . (3.1.16a) Here ξ1 > 0 and ξ2 > 0 are defined in terms of µ by ξ1 = √ 1 + √ 1 + 4µδ 2δ , ξ2 = √ −1 +√1 + 4µδ 2δ , (3.1.16b) where the eigenvalues µ are the roots of the transcendental equation 2ξ1 + ( ξ21 − ξ22 ξ2 ) sin ξ2 sinh ξ1 − 2ξ1 cosh ξ1 cos ξ2 = 0 . (3.1.16c) 31 3.2. Biharmonic Nonlinear Eigenvalue Problem: Slab Geometry Slab Unit Disc δ λ̄ λc λ̄ λc 0.25 20.3576 19.249 4.886 4.395 0.5 38.900 36.774 8.754 7.871 1.0 75.979 71.823 16.486 14.826 2.0 150.137 141.918 31.948 28.704 Table 3.1: Upper bounds, λ̄, for the fold point location given in (3.1.15) compared with the numerically computed fold point location λc of (1.1.4). Similarly, for the unit disk 0 < r < 1, the eigenfunctions are given up to a scalar multiple by φ = J0(ξ2r)− J0(ξ2) I0(ξ1) I0(ξ1r) , (3.1.17a) where J0 and I0 are the Bessel and modified Bessel functions of the first kind of order zero, respectively. The eigenvalues µ are the roots of the transcendental equation ξ1I1(ξ1) + ξ2 I0(ξ1) J0(ξ2) J1(ξ2) = 0 , (3.1.17b) where J1(ρ) = −J ′0(ρ) and I1(ρ) = I ′0(ρ). The first root of (3.1.16c) and (3.1.17b) corresponding to the principle eigenvalue, µ0, of (3.1.11) is readily computed using Newton’s method as a function of δ > 0. Then, the corresponding principal eigenfunction φ0 from either (3.1.16a) or (3.1.17a) can be readily verified numerically to have one sign on Ω. Then, in terms of µ0, (3.1.15) gives an explicit upper bound for λc. These bounds for λc together with the full numerical results for λc are compared in Table 3.1. From this table it is observed that the upper bound provides is actually relatively close to the true location for the fold point. 3.2 Biharmonic Nonlinear Eigenvalue Problem: Slab Geometry In this section an asymptotic expansion for the fold point at the end of the minimal solution branch for (1.1.4) is developed in a slab domain. This fold point determines the onset of the pull-in instability, and hence its determination is important in the actual design of a MEMS device. 32 3.2. Biharmonic Nonlinear Eigenvalue Problem: Slab Geometry In a slab domain (1.1.4) becomes − δu′′′′ + u′′ = λ (1 + u)2 , 0 < x < 1 ; u(0) = u(1) = u′(0) = u′(1) = 0 . (3.2.1) In the limit δ ≪ 1, (3.2.1) is a singular perturbation problem for which the solution u has boundary layers near both endpoints x = 0 and x = 1. The width of these boundary layers is found below to be O(δ1/2), which motivates an asymptotic expansion for the fold point location in powers of O(δ1/2). Therefore, in the outer region defined away from O(δ1/2) neighborhoods of both endpoints, u and λ are expanded as u = u0 + δ 1/2u1 + δu2 + · · · , λ = λ0 + δ1/2λ1 + δλ2 + · · · . (3.2.2) Upon substituting (3.2.2) into (3.2.1), and collecting powers of δ1/2, the following se- quence of problems is obtained u′′0 = λ0 (1 + u0)2 , 0 < x < 1 , (3.2.3a) Lu1 = λ1 (1 + u0)2 , 0 < x < 1 , (3.2.3b) Lu2 = λ2 (1 + u0)2 − 2λ1u1 (1 + u0)3 + 3λ0u 2 1 (1 + u0)4 + u′′′′0 , 0 < x < 1 . (3.2.3c) Here L is the linear operator defined by Lφ ≡ φ′′ + 2λ0 (1 + u0)3 φ . (3.2.4) Next, appropriate boundary conditions for u0, u1 and u2 as x → 0 and x → 1 are determined. These conditions are obtained by matching the outer solution to boundary layer solutions defined in the vicinity of x = 0 and x = 1. In the boundary layer region near x = 1, the following inner variables y and v(y) are introduced together with the inner expansion for v; y = δ−1/2(x− 1) , u = δ1/2v , v = v0 + δ1/2v1 + δv2 + · · · . (3.2.5) Upon substitution of (3.2.5) and (3.2.2) for λ into (3.2.1), powers of δ1/2 are collected to obtain on −∞ < y < 0 that −v′′′′0 + v′′0 = 0 , v0(0) = v′0(0) = 0 , (3.2.6a) −v′′′′1 + v′′1 = λ0 , v1(0) = v′1(0) = 0 , (3.2.6b) −v′′′′2 + v′′2 = λ1 − 2λ0v0 , v2(0) = v′2(0) = 0 . (3.2.6c) 33 3.2. Biharmonic Nonlinear Eigenvalue Problem: Slab Geometry The solution to (3.2.6) with no exponential growth as y → −∞ is given in terms of unknown constants c0, c1, and c2, by v0 = c0 (−1− y + ey) , (3.2.7a) v1 = c1 (−1− y + ey) + λ0y2/2 , (3.2.7b) v2 = c2 (−1− y + ey) + λ1y2/2 + c0λ0y (−1 + y + ey + y2/3) . (3.2.7c) The matching condition to the outer solution is obtained by letting y → −∞ and substituting v0, v1, and v2 into (3.2.5) for u, and then writing the resulting expression in terms of outer variables. In this way, the following matching condition as x → 1 is established: u ∼ −c0(x− 1) +O((x− 1)2) + δ1/2 [−c0 − c1(x− 1) +O((x− 1)2)] +δ [−c1 − (c0λ0 + c2)(x− 1) +O((x− 1)2)]+ · · · . (3.2.8) This matching condition not only gives appropriate boundary conditions to the outer problems for u0, u1, and u2, defined in (3.2.3), but it also determines the unknown constants c0, c1, and c2 in the inner solutions (3.2.7) in a recursive way. In particular, the O(δ0) term in (3.2.8) yields that u0 = 0 at x = 1 and that c0 is then given by c0 = −u′0(1). The remaining O(δ0) terms in (3.2.8) then match identically as seen by using the solution u0 to (3.2.3a). In a similar way, boundary conditions for u1 and u2 and formulae for the constants c1 and c2 are established. A similar analysis can be performed for the boundary layer region at the other endpoint x = 0. This analysis is identical to that near x = 1 since u0, u1 and u2 are symmetric about the mid-line x = 1/2. In this way, the following boundary conditions for (3.2.3) are obtained: u0(0) = u0(1) = 0 , u1(0) = u1(1) = u ′ 0(1) , u2(0) = u2(1) = u ′ 1(1) . (3.2.9) The constants c0, c1, and c2, in (3.2.7) that are associated with the boundary layer solution near x = 1 are given by c0 = −u′0(1) , c1 = −u′1(1) , c2 = −u′2(1) + λ0u′0(1) , (3.2.10) which then determines the boundary layer solution in (3.2.7) explicitly. Therefore, (3.2.3) for u0, u1, and u2, must be solved subject to the boundary condi- tions as given in (3.2.9). With the introduction of α = u0(1/2), a parameterization of the minimal solution branch for u0 and λ0 is established and the dependence uj = uj(x, α) for j = 0, 1, 2 follows. It is readily verified that the solution to (3.2.3b) is given by (see 34 3.2. Biharmonic Nonlinear Eigenvalue Problem: Slab Geometry Lemma 3.2 below) u1 = λ1 3λ0 (1 + u0) , (3.2.11) where λ1 is found by satisfying u1(1) = u ′ 0(1). Therefore, for δ ≪ 1, the following explicit two-term expansions for both the outer solution and for the global bifurcation curve λ(α) is developed: u ∼ u0(x;α) + δ1/2u′0(1, α) [1 + u0(x, α)] +O(δ) , λ ∼ λ0(α) + 3λ0(α)u′0(1, α)δ1/2 +O(δ) . (3.2.12) It is noted that this “global” perturbation result for λ is not uniformly valid in the limit α → −1 corresponding to λ0 → 0. In this limit, the term (1 + u0)−2 is nearly singular at x = 1/2, and a different asymptotic analysis is required (see § 5 of [30] and Chapter (4) of this work). A higher-order local analysis of the bifurcation diagram near the fold point on the minimal solution branch is now constructed. This minimal solution branch for u0 is well- known to have a fold point at α = α0 ≈ −0.389 at which λc ≡ λ0(α0) ≈ 1.400. This point determines the pull-in voltage for the unperturbed problem. To determine the location of the fold point for the perturbed problem, expand α(δ) = α0 + δ 1/2α1 + δα2, where αj is determined by the condition that dλ/dα = 0 is independent of δ. Defining λc(δ) = λ(α(δ), δ), the expansion of the fold point for (3.2.1) when δ ≪ 1 is determined to be λc = λ0c + δ 1/2λ1(α0) + δ [ λ2(α0)− λ 2 1α(α0) 2λ0αα(α0) ] +O(δ3/2) . (3.2.13) Here the subscript indicates derivatives in α. Therefore, to determine a three-term expansion for the fold point as in (3.2.13), the quantities λ1, λ2, λ1α and λ0αα must be calculated at the unperturbed fold point location α0 from the solution to (3.2.3) with boundary conditions (3.2.9). To do so, the problems for u0 and u1 in (3.2.3) are first differentiated with respect to α to obtain on 0 < x < 1 that Lu0α = λ0α (1 + u0)2 , (3.2.14a) Lu0αα = λ0αα (1 + u0)2 − 4λ0αu0α (1 + u0)3 + 6λ0u 2 0α (1 + u0)4 , (3.2.14b) Lu1α = λ1α (1 + u0)2 − 2λ1u0α (1 + u0)3 − 2λ0αu1 (1 + u0)3 + 6λ0u1u0α (1 + u0)4 . (3.2.14c) Here L is the linear operator defined in (3.2.4). At the unperturbed fold location 35 3.2. Biharmonic Nonlinear Eigenvalue Problem: Slab Geometry α = α0, where λ0α = 0, the function u0α is a nontrivial solution satisfying Lu0α = 0. Therefore, λ1(α0), λ2(α0), λ0αα(α0), and λ1α(α0), can be calculated by applying a Fredholm solvability condition to each of (3.2.3b), (3.2.3c), (3.2.14b), and (3.2.14c), respectively. Upon applying Lagrange’s identity to (3.2.14a) and (3.2.3b) at α = α0, the following equality is established: ∫ 1 0 u0αLu1 dx = ∫ 1 0 u1Lu0α dx = −u1(1)u′0α(1) + u1(0)u′0α(0) . Therefore, since u1(1) = u1(0) = u ′ 0(1) from (3.2.9), and u ′ 0α(1) = −u′0α(0), it follows at α = α0 that λ1I = −2u′0(1)u′0α(1) , I ≡ ∫ 1 0 u0α (1 + u0)2 dx . (3.2.15) The integral I can be evaluated more readily using the following lemma: Lemma 3.1: At α = α0, the following identity holds: I ≡ ∫ 1 0 u0α (1 + u0)2 dx = − 2 3λ0 u′0α(1) . (3.2.16) To prove this lemma, (3.2.14a) is first multiplied by (1 + u0) and then integrated over 0 < x < 1. Setting α = α0, integrating by parts twice, and then using u ′′ 0 = λ0/(1 + u0) 2, results in the following sequence of equalities: I = − 1 2λ0 ∫ 1 0 (1 + u0)u ′′ 0α dx = − 1 2λ0 [ 2u′0α(1) + ∫ 1 0 u0αu ′′ 0 dx ] = −u ′ 0α(1) λ0 − 1 2 ∫ 1 0 u0α (1 + u0)2 dx . This last expression gives I = −u′0α(1)/λ0 − I/2 which is rearranged to yield (3.2.16), and completes the proof of Lemma 3.1. Next, (3.2.16) is substituted into (3.2.15) and evaluated at α = α0 to reveal that λ1 = 3λ0u ′ 0(1) . (3.2.17) This result is consistent with the global perturbation result (3.2.12) when it is evaluated at α = α0. The values of λ0αα, λ1α, and λ0αα, at α = α0 can be evaluated by imposing similar solvability conditions with respect to u0α. From (3.2.14a) and (3.2.14b), and by using 36 3.2. Biharmonic Nonlinear Eigenvalue Problem: Slab Geometry (3.2.16) for I, it is readily shown at α = α0 that λ0αα = 9λ20 u′0α(1) ∫ 1 0 u30α (1 + u0)4 dx . (3.2.18) Next, from (3.2.14a) and (3.2.14c), we calculate at α = α0 that ∫ 1 0 u0αLu1α dx = −u1αu′0α ∣∣∣1 0 = −2u1α(1)u′0α(1) . Upon using (3.2.14c) for Lu1α and u1α(1) = u′0α(1) from (3.2.9), the expression above becomes λ1αI = − ∫ 1 0 u0α ( 6λ0u1u0α (1 + u0)4 − 2λ1u0α (1 + u0)3 ) dx− 2 [u′0α(1)]2 . (3.2.19) In a similar way, λ2 is evaluated at α = α0 by application of Lagrange’s identity to (3.2.14a) and (3.2.3c) to obtain λ2I = − ∫ 1 0 u0α [ 3λ0u 2 1 (1 + u0)4 − 2λ1u1 (1 + u0)3 + u′′′′0 ] dx− 2u′0α(1)u′1(1) . (3.2.20) The formulae above for λ1α and λ2 at α = α0, which are needed in (3.2.13), can be simplified considerably by using the following simple result: Lemma 3.2: At α = α0, the solution u1 to (3.2.3b) with u1(1) = u1(0) = u ′ 0(1) is given, for any constant D, by u1 = λ1 3λ0 (1 + u0) +Du0α . (3.2.21) Moreover, the correction term of order O(δ) in the expansion (3.2.13) of the fold point is independent of D. The proof is by a direct calculation. Clearly u1 solves (3.2.3b) at α = α0 since Lu1 = λ1 3λ0 L(1 + u0) = λ1 3λ0 [ u′′0 + 2λ0 (1 + u0)2 ] = λ1 3λ0 [ λ0 (1 + u0)2 + 2λ0 (1 + u0)2 ] = λ1 (1 + u0)2 . In addition, since u0(1) = 0, then u1(1) = λ1/3λ0 = u ′ 0(1) from (3.2.17), as re- quired by (3.2.9). Finally, a tedious but direct computation using (3.2.18), (3.2.19), and (3.2.20), shows that λ2 − λ21α/[2λ0αα] at α = α0 is independent of the constant D in (3.2.21). Therefore, the fold point correction is independent of the normalization of u1. The details of this latter calculation are left to the reader. 37 3.2. Biharmonic Nonlinear Eigenvalue Problem: Slab Geometry Therefore, D = 0 is taken to get u1 = λ1(1+u0)/(3λ0). Upon substitution of u1 into (3.2.19), it is observed that the integral term on the right-hand side of (3.2.19) vanishes identically. Then, using (3.2.16) for I, the following compact formula is obtained at α = α0: λ1α = 3λ0u ′ 0α(1) . (3.2.22) Reassuringly, this agrees with differentiation of (3.2.12) by α followed by evaluation at α0. Similarly, in (3.2.20) for λ2, one sets u1 = λ1(1 + u0)/(3λ0) and u2(1) = u ′ 1(1) = λ1u ′ 0(1)/(3λ0), to obtain λ2I = −2λ1 3λ0 u′0(1)u ′ 0α(1) + λ21 3λ0 I − ∫ 1 0 u0αu ′′′′ 0 dx . (3.2.23) Expression (3.2.23) can be reduced further by integrating twice by parts as follows: ∫ 1 0 u0αu ′′′′ 0 dx = − ∫ 1 0 u′0αu ′′′ 0 dx = −u′0αu′′0 ∣∣∣1 0 + ∫ 1 0 u′′0αu ′′ 0 dx = −2u′0α(1)λ0 + ∫ 1 0 ( − 2λ0u0α (1 + u0)3 )( λ0 (1 + u0)2 ) dx = −2λ0u′0α(1)− 2λ20 ∫ 1 0 u0α (1 + u0)5 dx . Combining this last expression with (3.2.23) together with the formula for I in (3.2.16) and λ1 = 3λ0u ′ 0(1), it follows at α = α0 that λ2 = 6λ0 [ u′0(1) ]2 − 3λ20 − 3λ30u′0α(1) ∫ 1 0 u0α (1 + u0)5 dx . (3.2.24) The results of the preceding calculations are summarized in the following statement: Principal Result 3.3: Let α0, λ0c ≡ λ0(α0) be the location of the fold point at the end of the minimal solution branch for (3.2.3a) with boundary conditions u0(0) = u0(1) = 0. Then, for the singularly perturbed problem (3.2.1), a three-term expansion for the perturbed fold point location is λc = λ0c + 3λ0δ 1/2u′0(1) + δλ̂2 + · · · , λ̂2 ≡ λ2(α0)− λ21α(α0) 2λ0αα(α0) . (3.2.25a) Here λ̂2 is defined in terms of u0 and u0α by λ̂2 = 6λ0 [ u′0(1) ]2 − 3λ20 − 3λ30u′0α(1) ∫ 1 0 u0α (1 + u0)5 dx− [u ′ 0α(1)] 3 2 (∫ 1 0 u30α (1 + u0)4 dx )−1 . (3.2.25b) 38 3.3. Biharmonic Nonlinear Eigenvalue Problem: Multidimensional Domain For the unit slab, the minimal solution branch for the unperturbed problem (3.2.3a) can be obtain implicitly in terms of the parameter α ≡ u0(1/2). Multiply (3.2.3a) by u′0 and integrate once to obtain u ′ 0 = √ 2λ0 1 + α ( u− α u+ 1 )1/2 . (3.2.26) A further integration using u0 (1/2) = α and u0(1) = 0, determines λ0(α) as λ0(α) = 2(1 + α) [ 2 ∫ 1 √ 1+α s2 ds√ s2 − (1 + α) ]2 = 2(1 + α) [√−α+ (1 + α) log(1 +√−α√ 1 + α )]2 . (3.2.27) Upon setting λ0α = 0, the fold point α0 ≈ −0.389 and λ0c ≈ 1.400 is determined. By using this solution the various terms needed in (3.2.25) are easily calculated. In this way, (3.2.25) leads to the following explicit three-term expansion valid for δ ≪ 1: λc = 1.400 + 5.600 δ 1/2 + 25.451 δ + · · · . (3.2.28) In Fig. 3.5(b) a comparison of the two-term and three-term asymptotic results for λc versus δ from (3.2.28) is provided alongside the corresponding full numerical result computed from (1.1.4). The three-term approximation in (3.2.28) is seen to provide a reasonably accurate determination of λc. For δ = 0.01, in Fig. 3.5(a) the two-term approximation (3.2.12) is compared with the global bifurcation curve with the full nu- merical result computed from (1.1.4) and from the membrane MEMS problem (1.1.2), corresponding to δ = 0. It is clear that the fold point location depends rather sensitively on δ, even when δ ≪ 1, owing to the O(δ1/2) limiting behaviour. 3.3 Biharmonic Nonlinear Eigenvalue Problem: Multidimensional Domain The results of § 3.2 are now extended to the case of a bounded two-dimensional domain Ω with smooth boundary ∂Ω. Equation (1.1.4) is considered in the limit δ → 0, and it is assumed that the fold point location at the end of the minimal solution branch u0(x, α), λ0(α) for the unperturbed problem ∆u0 = λ0 (1 + u0)2 , x ∈ Ω ; u0 = 0 , x ∈ ∂Ω , (3.3.1) 39 3.3. Biharmonic Nonlinear Eigenvalue Problem: Multidimensional Domain 1.0 0.8 0.6 0.4 0.2 0.0 2.52.01.51.00.50.0 |α| λ (a) |α| vs. λ (Unit Slab) 4.0 3.0 2.0 1.0 0.0 0.030.020.010.00 λc δ (b) λc vs. δ (Unit Slab) Figure 3.5: Left figure: Plot of numerically computed global bifurcation diagram |α| = |u (1/2) | versus λ for (1.1.4) with δ = 0.01 (heavy solid curve) compared to the two-term asymptotic result (3.2.12) (solid curve) and the unperturbed δ = 0 membrane MEMS result from (1.1.2) (dashed curve). Right figure: Comparison of numerically computed fold point λc versus δ (heavy solid curve) with the two-term (dashed curve) and the three-term (solid curve) asymptotic result from (3.2.28). has been determined. This fold point location is labeled as λ0c = λ0(α0) for some α = α0. For an arbitrary domain Ω, α can be chosen to be the L2 norm of u0. For the unit disk, where u0(r) is radially symmetric, it is more convenient to define α by α = u0(0). For the perturbed problem (1.1.4), λ and the outer solution for u are expanded in powers of δ1/2 as in (3.2.2), to obtain Lu1 ≡ ∆u1 + 2λ0 (1 + u0)3 u1 = λ1 (1 + u0)2 , x ∈ Ω , (3.3.2a) Lu2 = λ2 (1 + u0)2 − 2λ1u1 (1 + u0)3 + 3λ0u 2 1 (1 + u0)4 +∆2u0 , x ∈ Ω . (3.3.2b) The expansion of the perturbed fold point location is again as given in (3.2.13). To derive boundary conditions for u1 and u2, a boundary layer solution near ∂Ω with width O(δ1/2) is constructed. It is advantageous to implement an orthogonal coordinate system η, s, where η > 0 measures the perpendicular distance from x ∈ Ω to ∂Ω, whereas on ∂Ω the coordinate s denotes arclength. In terms of (η, s), (1.1.4) transforms to −δ ( ∂ηη − κ 1− κη∂η + 1 1− κη∂s ( 1 1− κη∂s ))2 u + ( ∂ηηu− κ 1− κη∂ηu+ 1 1− κη∂s ( 1 1− κη∂su )) = λ (1 + u)2 . (3.3.3) 40 3.3. Biharmonic Nonlinear Eigenvalue Problem: Multidimensional Domain Here κ = κ(s) is the curvature of ∂Ω, with κ = 1 for the unit disk. The inner variables and the inner expansion, defined in an O(δ1/2) neighborhood of ∂Ω, are then introduced as η̂ = η/δ1/2 , u = δ1/2v , v = v0 + δ 1/2v1 + δv2 + · · · . (3.3.4) After substituting (3.3.4) into (3.3.3) and collecting powers of δ, some lengthy but straightforward algebra produces the following sequence of problems on −∞ < η̂ < 0: −v0η̂η̂η̂η̂ + v0η̂η̂ = 0 , v0 = v0η̂ = 0 , on η̂ = 0 ; (3.3.5a) −v1η̂η̂η̂η̂ + v1η̂η̂ = −2κv0η̂η̂η̂ + κv0η̂ + λ0 , v1 = v1η̂ = 0 , on η̂ = 0 , (3.3.5b) −v2η̂η̂η̂η̂ + v2η̂η̂ = −2κv1η̂η̂η̂ + κv1η̂ − 2κ2η̂v0η̂η̂η̂ − κ2v0η̂η̂ + κ2η̂v0η̂ + 2v0η̂η̂ss − v0ss + λ1 − 2λ0v0 , v2 = v2η̂ = 0 , on η̂ = 0 . (3.3.5c) The asymptotic behaviour of the solution to (3.3.5) with no exponential growth as η → −∞ is given in terms of unknown functions c0(s), c1(s), and c2(s) by v0 ∼ −c0 + c0η̂ , v1 ∼ −c1 + ( c1 − c0κ 2 ) η̂ , with v2 ∼ −c2+O(η) as η → 0. Therefore, with u = δ1/2v, and by rewriting v in terms of the outer variable η = η̂δ1/2, the following matching condition, analogous to (3.2.8), is obtained for the outer solution: u ∼ c0η + δ1/2 [ −c0 + η ( c1 − c0κ 2 )] + δ [−c1 +O(η)] + · · · . (3.3.6) Noting that the outer normal derivative ∂nu on ∂Ω is simply ∂nu = −∂ηu, (3.3.6) then implies the following boundary conditions for the outer solutions u1 and u2 in (3.3.2): u0 = 0 , u1 = ∂nu0 , u2 = ∂nu1 + κ 2 ∂nu0 , x ∈ ∂Ω . (3.3.7) The functions c0(s) and c1(s), which determine the leading two boundary layers solu- tions explicitly, are given by c0 = −∂nu0 , c1 = −∂nu1 − κ 2 ∂nu0 , x ∈ ∂Ω , with a more complicated expression, which we omit, for c2(s). Notice that the boundary condition for u2 on ∂Ω depends on the curvature κ of ∂Ω. The remainder of the analysis to calculate the terms in the expansion of the fold point is similar to that in §3. At α = α0, Lu0α = 0, and so each of the problems in (3.3.2) must satisfy a solvability condition. By applying Green’s identity to u0α and u1, 41 3.3. Biharmonic Nonlinear Eigenvalue Problem: Multidimensional Domain together with the boundary condition u1 = ∂nu0 on ∂Ω, it follows at α = α0 that λ1I = − ∫ ∂Ω (∂nu0) (∂nu0α) dx , I ≡ ∫ Ω u0α (1 + u0)2 dx . (3.3.8) The integral I can be written more conveniently by using the following lemma: Lemma 3.4: At α = α0, the following identity holds: I ≡ ∫ Ω u0α (1 + u0)2 dx = − 1 3λ0 ∫ ∂Ω ∂nu0α dx. (3.3.9) To prove this result, the equation for u0α together with Green’s second identity and the divergence theorem is used to calculate I = − 1 2λ0 ∫ Ω (1 + u0)∆u0α dx = − 1 2λ0 [∫ ∂Ω ∂nu0α dx+ ∫ Ω u0α∆u0 dx ] = − 1 2λ0 ∫ ∂Ω ∂nu0α dx− I 2 . Solving for I then gives the result. Upon substituting (3.3.9) into (3.3.8), λ1 can be expressed at α = α0 as λ1 = 3λ0 (∫ ∂Ω (∂nu0) (∂nu0α) dx∫ ∂Ω ∂nu0α dx ) . (3.3.10) From (3.2.13) this then specifies the correction of orderO(δ1/2) to the fold point location. To determine the O(δ) term in the expansion (3.2.13) of the fold point, the terms λ0αα, λ1α, and λ2 at α = α0 must be calculated. This is done through solvability conditions with u0α in a similar way as in § 3.2. This procedure leads to the following identities at α = α0: λ0αα = 18λ20(∫ ∂Ω ∂nu0α dx ) ∫ Ω u30α (1 + u0)4 dx , (3.3.11a) λ1αI = − ∫ Ω u0α ( 6λ0u1u0α (1 + u0)4 − 2λ1u0α (1 + u0)3 ) dx− ∫ ∂Ω [∂nu0α] 2 dx , (3.3.11b) λ2I = − ∫ Ω u0α [ 3λ0u 2 1 (1 + u0)4 − 2λ1u1 (1 + u0)3 +∆2u0 ] dx− ∫ ∂Ω [ ∂nu1 + κ 2 ∂nu0 ] ∂nu0α dx . (3.3.11c) In contrast to the one-dimensional case of § 3.2, u1 cannot be obtained as explicitly as in Lemma 3.2. In place of Lemma 3.2, it is readily shown that u1 admits the following 42 3.3. Biharmonic Nonlinear Eigenvalue Problem: Multidimensional Domain decomposition at α = α0: u1 = λ1 3λ0 (1 + u0) + u1a +Du0α . (3.3.12) Here D is any scalar constant, and u1a at α = α0 is the unique solution to Lu1a = 0 , x ∈ Ω ; u1a = ∂nu0 − λ1 3λ0 , x ∈ ∂Ω ; ∫ Ω u1au0α dx = 0 . (3.3.13) By substituting (3.3.12) into (3.3.11), a straightforward calculation shows that the quan- tity λ2 − λ21α/[2λ0αα] is independent of D. Hence, set D = 0 in (3.3.12) for simplicity. By substituting (3.3.12) for u1 into (3.3.11b), and then using (3.3.9) for I, λ1α at α = α0 can be written as λ1α = 18λ20(∫ ∂Ω ∂nu0α dx ) ∫ Ω u1au 3 0α (1 + u0)4 dx+ 3λ0 (∫ ∂Ω [∂nu0α] 2 dx∫ ∂Ω ∂nu0α dx ) . (3.3.14) Similarly, equation (3.3.12) for u1 and (3.3.9) for I are substituted into (3.3.11c). In addition, in the resulting expression, the following identity which is readily derived by integration by parts, is used:∫ Ω u0α∆ 2u0 dx = −2λ20 ∫ Ω u0α (1 + u0)5 dx− λ0 ∫ ∂Ω ∂nu0α dx . (3.3.15) In this way, expression (3.3.11c) for λ2 at α = α0 simplifies to λ2 = 2λ21 3λ0 − 3λ20 + 3λ0(∫ ∂Ω ∂nu0α dx ) ∫ ∂Ω [ ∂nu1a + κ 2 ∂nu0 ] ∂nu0α dx − 6λ 3 0(∫ ∂Ω ∂nu0α dx ) ∫ Ω u0α (1 + u0)5 dx+ 9λ20(∫ ∂Ω ∂nu0α dx ) ∫ Ω u21au0α (1 + u0)4 dx . (3.3.16) The results of the preceding calculations are summarized as follows: Principal Result 3.5: Let λc ≡ λ0(α0) be the fold point location at the end of the min- imal solution branch for the unperturbed problem (3.3.1) in a bounded two-dimensional domain Ω, with smooth boundary ∂Ω. Then, for (1.1.4) with δ ≪ 1, a three-term expansion for the perturbed fold point location is λc = λ0c + 3λ0δ 1/2 (∫ ∂Ω (∂nu0) (∂nu0α) dx∫ ∂Ω ∂nu0α dx ) + δλ̂2 + · · · , λ̂2 ≡ λ2(α0)− λ 2 1α(α0) 2λ0αα(α0) . (3.3.17) Here λ0αα(α0), λ1α(α0), and λ2(α0) are as given in (3.3.11a), (3.3.14), and (3.3.16), respectively. The form of the δ∞/∈ correction term in (3.3.17) of Principal Result 3.5 may give 43 3.3. Biharmonic Nonlinear Eigenvalue Problem: Multidimensional Domain some insight into the relationship between the curvature of the domain boundary and the principal fold point λc of (1.1.4). Specifically, a boundary with large curvature generates a larger perimeter |∂Ω| and as ∂nu0 is positive on ∂Ω, one expects that the value of the integral term is larger for domains with large curvatures. For the special case of the unit disk where Ω := {x | |x| ≤ 1}, then u0 = u0(r) and u0α = u0α(r) are radially symmetric, and κ = 1. Therefore, for this special geometry, u1a ≡ 0 from (3.3.13), and consequently the various terms in (3.3.17) can be simplified considerably. In analogy with Principal Result 3.3, the following asymptotic expansion is obtained for the fold point location of (1.1.4) in the limit δ → 0 for the unit disk: Corollary 3.6: For the special case of the unit disk, let α0 = u0(0) and λ0c ≡ λ0(α0) be the location of the fold point at the end of the minimal radially symmetric solution branch for the unperturbed problem (3.3.1). Then, for (1.1.4) with δ ≪ 1, a three-term expansion for the perturbed fold point location is λc = λ0c + 3λ0δ 1/2u′0(1) + δλ̂2 + · · · , λ̂2 ≡ λ2(α0)− λ21α(α0) 2λ0αα(α0) . (3.3.18a) Here λ̂2 is defined in terms of u0 and u0α by λ̂2 = 3 2 λ0u ′ 0(1) + 6λ0 [ u′0(1) ]2 − 3λ20 − 6λ 3 0 u′0α(1) ∫ 1 0 ru0α (1 + u0)5 dr − [u ′ 0α(1)] 3 2 (∫ 1 0 ru30α (1 + u0)4 dr )−1 . (3.3.18b) The first term in λ̂2 above arises from the constant curvature of ∂Ω. For the unit disk, numerical values for the coefficients in the expansion (3.3.18) are obtained by first using COLSYS [1] to solve for u0 and u0α. In this way, the explicit three-term expansion for the unit disk is λc = 0.789 + 1.578 δ 1/2 + 6.261 δ + · · · . (3.3.19) In addition, for the unit disk it follows as in (3.2.12) that the global bifurcation diagram is given for δ ≪ 1 by λ ∼ λ0(α) + 3λ0(α)u′0(1, α)δ1/2 +O(δ) . (3.3.20) For the unit disk, Fig. 3.6(b) provides a comparison of the two-term and three- term asymptotic results for λc versus δ from (3.3.19) along with the corresponding full numerical result computed from (1.1.4). Since the coefficients in (3.3.19) are smaller than those in (3.2.28), the three-term approximation for the unit disk is seen to provide a more accurate determination of λc than the result for the slab shown in Fig. 3.5(b). 44 3.4. Perturbing from the Pure Biharmonic Eigenvalue Problem For δ = 0.01, Fig. 3.6(a) provides a comparison of the two-term approximation (3.3.20) to the global bifurcation curve with the full numerical result computed from (1.1.4) and also the membrane MEMS problem (1.1.2), corresponding to δ = 0. From this figure it is seen that (3.3.20) compares favorably with the full numerical result provided that α is not too close to −1. Recall that λ → 4/9 as α → −1 for (1.1.2), whereas from the numerical results in §2, λ → 0 as α → −1 for (1.1.4) when δ > 0. The singular limit α→ −1 is examined in Chapter 4. 1.0 0.8 0.6 0.4 0.2 0.0 1.21.00.80.60.40.20.0 |α| λ (a) |α| vs. λ (Unit Disk) 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.030.020.010.00 λc δ (b) λc vs. δ (Unit Disk) Figure 3.6: Left figure: Plot of numerically computed global bifurcation diagram |α| = |u (0) | versus λ for (1.1.4) with δ = 0.01 (heavy solid curve) compared to the two-term asymptotic result (3.3.20) (solid curve) and the unperturbed δ = 0 membrane MEMS result from (1.1.2) (dashed curve). Right figure: Comparison of numerically computed fold point λc versus δ (heavy solid curve) with the two-term (dashed curve) and the three-term (solid curve) asymptotic result from (3.3.19). 3.4 Perturbing from the Pure Biharmonic Eigenvalue Problem Next, equation (1.1.4) is considered in the limit δ ≫ 1. To study this limiting case, (1.1.4) is rewritten as −∆2u+ 1 δ ∆u = λ̃ (1 + u)2 , x ∈ Ω ; u = ∂nu = 0 , x ∈ ∂Ω , (3.4.1) where λ̃ ≡ λ/δ. For δ ≫ 1, the solution u and the nonlinear eigenvalue parameter λ̃ are expanded as u = u0 + 1 δ u1 + · · · , λ̃ = λ̃0 + 1 δ λ̃1 + · · · . (3.4.2) 45 3.4. Perturbing from the Pure Biharmonic Eigenvalue Problem Inserting this expansion in (3.4.1) and collecting terms, requires that u0 satisfies −∆2u0 = λ̃0 (1 + u0)2 , x ∈ Ω ; u0 = ∂nu0 = 0 , x ∈ ∂Ω , (3.4.3a) and that u1 satisfies Lbu1 ≡ ∆2u1− 2λ̃0 (1 + u0)3 u1 = ∆u0− λ̃1 (1 + u0)2 , x ∈ Ω; u1 = ∂nu1 = 0 , x ∈ ∂Ω. (3.4.3b) The minimal solution branch for the unperturbed problem (3.4.3a) is parameterized as λ̃0(α), u0(x, α). It is assumed that there is a simple fold point at the end of this branch with location α = α0, where λ̃0c = λ̃0(α0). Since Lbu0α = 0 at α = α0, the solvability condition for (3.4.3b) determines λ̃1 at α = α0 as Ibλ̃1 = ∫ Ω u0α∆u0 dx , Ib = ∫ Ω u0α (1 + u0)2 dx . (3.4.4) By using the equation and boundary conditions for u0α and u0, we can calculate Ib using Green’s identity as Ib = 1 2λ0 ∫ Ω (u0 + 1)∆ 2u0α dx = 1 2λ0 [∫ ∂Ω ∂n (∆u0α) dx+ ∫ Ω u0α∆ 2u0 dx ] = 1 2λ0 ∫ ∂Ω ∂n (∆u0α) dx− Ib 2 . (3.4.5) This yields that Ib = 1 3λ0 ∫ ∂Ω ∂n (∆u0α) dx . (3.4.6) Principal Result 3.7: Let λ̃0(α) and u0(x, α) be the minimal solution branch for the pure biharmonic problem (3.4.3a), and assume that there is a simple fold point at α = α0 where λ̃0c = λ̃0(α0). Then, for δ ≫ 1, the expansion of the fold point for (1.1.4) is given by λc ∼ δ [ λ̃0c + δ −1λ̃1(α0) +O(δ−2) ] , λ̃1(α0) ≡ 3λ0 ( ∫ Ω u0α∆u0 dx∫ ∂Ω ∂n (∆u0α) dx ) . (3.4.7a) For the special case of the unit disk or a slab domain of unit length, then λ̃1(α0) reduces to λ̃1(α0) = 3λ0 ∂r (∆u0α) ∣∣∣ r=1 ∫ 1 0 u0α (ru0r)r dr , (Unit Disk) ; λ̃1(α0) = 3λ0 2u′′′0α(1) ∫ 1 0 u0αu ′′ 0 dx (Unit Slab) . (3.4.7b) 46 3.5. The Fringing-Field and Annulus Problems For the slab and the unit disk, the numerical values of the coefficients in the ex- pansion (3.4.7) are calculated by using COLSYS [1] to solve for u0 and u0α at the fold point of the minimal branch for the pure Biharmomic problem (3.4.3a). In this way, the following is obtained for δ ≫ 1: λc ∼ δ [ 70.095 + 1 δ 1.729 + · · · ] (Unit Slab) ; λc ∼ δ [ 15.412 + 1 δ 1.001 + · · · ] (Unit Disk) . (3.4.8) Although (3.4.8) was derived in the limit δ ≫ 1, in Fig. 3.7 it is shown, rather remarkably, that (3.4.8) is also in rather close agreement with the full numerical result for λc, computed from (1.1.4), even when δ < 1. Therefore, for the unit disk and the unit slab, the limiting approximations for λc when δ ≪ 1 from (3.2.28) and (3.3.19), together with the δ ≫ 1 result (3.4.8), can be used to rather accurately predict the fold point λc for (1.1.4) for a wide range of δ > 0. 5.0 4.0 3.0 2.0 1.0 0.0 0.040.030.020.010.00 λc δ (a) λc vs. δ (Unit Slab) 5.0 4.0 3.0 2.0 1.0 0.0 0.250.200.150.100.050.00 λc δ (b) λc vs. δ (Unit Disk) Figure 3.7: Comparison of full numerical result for λc versus δ (heavy solid curves) computed from (1.1.4) with the two-term asymptotic results (3.4.8) (solid curves) for the unit slab (left figure) and the unit disk (right figure). 3.5 The Fringing-Field and Annulus Problems 3.5.1 Fringing-Field Problem Next, the fringing-field problem (1.1.5) is considered in the limit δ → 0 in a two- dimensional domain Ω with smooth boundary ∂Ω. Let α0, λ0c = λ0(α0) be the location of the fold point for the unperturbed problem (3.3.1). To determine the leading order correction to the fold point location for (1.1.5) when δ ≪ 1, the solution to (1.1.5) is 47 3.5. The Fringing-Field and Annulus Problems expanded along the minimal solution branch as u = u0 + δu1 + · · · , λ = λ0 + δλ1 + · · · . (3.5.1) The problem for u1 is obtained by substituting (3.5.1) into (1.1.5), which yields Lu1 ≡ ∆u1 + 2λ0 (1 + u0)3 u1 = λ1 (1 + u0)2 + λ0 |∇u0|2 (1 + u0)2 , x ∈ Ω ; u1 = 0 x ∈ ∂Ω . (3.5.2) Since Lu0α = 0 at α = α0, the solvability condition for (3.5.2) at α = α0 determines λ1 at α = α0 as λ1(α0) = −λ0 I ∫ Ω |∇u0|2u0α (1 + u0)2 dx , I ≡ ∫ Ω u0α (1 + u0)2 dx , (3.5.3) where I is given in Lemma 4.1. For the special case of the unit disk and the unit slab the result is summarized as follows: Principal Result 3.8: Consider the fringing-field problem (1.1.5) with δ ≪ 1, and let λ0c be the fold point location for the unperturbed problem (3.3.1). Then, for δ ≪ 1, the fold point for the fringing-field problem (1.1.5) for the unit disk and a slab domain of unit length satisfies λc ∼ λ0c + 3λ 2 0δ u′0α(1) ∫ 1 0 ru20ru0α (1 + u0)2 dr , (Unit Disk) ; λc ∼ λ0c + 3λ 2 0δ 2u′0α(1) ∫ 1 0 (u′0) 2u0α (1 + u0)2 dx . (Unit Slab) . (3.5.4) For the special case of the unit slab, as well as the unit disk considered in [38], the coefficient in (3.5.4) can be evaluated from the numerical solution of (3.3.1) to obtain λc ∼ 0.789 − 0.160δ , (Unit Disk) ; λc ∼ 1.400 − 0.529δ , (Unit Slab) ; (3.5.5) Since the coefficients of δ are negative, the fringing field reduces the pull-in voltage. In Fig. 3.8(a) a very favorable comparison is shown between (3.5.5) for λc in the unit disk and the full numerical result computed from (1.1.5). A similar comparison for the unit slab is shown in Fig. 3.8(b). Since the coefficients of δ in (3.5.5) are rather small, (3.5.5) provides a rather good prediction of λc even for only moderately small δ. 3.5.2 The Annulus Problem Finally, the annulus problem (1.1.6) is considered in the limit δ ≪ 1 of a small inner radius. In this limit, (1.1.6) is singularly perturbed, and the construction of the solution 48 3.5. The Fringing-Field and Annulus Problems 0.80 0.78 0.76 0.74 0.72 0.70 0.50.40.30.20.10.0 λc δ (a) λc vs. δ (Unit Disk) 1.5 1.4 1.3 1.2 1.1 0.50.40.30.20.10.0 λc δ (b) λc vs. δ (Unit Slab) Figure 3.8: Left figure: Comparison of full numerical result for λc versus δ (heavy solid curve) for the fringing-field problem computed from (1.1.5) with the two-term asymp- totic result (3.5.5) (solid curve) for the unit disk. Right figure: A similar comparison for the unit slab. requires the matching of an outer solution defined for r = O(1) to an appropriate inner solution defined near r = δ ≪ 1. Related nonlinear eigenvalue problems for Bratu’s equation have been considered previously in [40] and [41]. Therefore, only a rather brief outline of the singular perturbation analysis is given here. Let u0(r, α), λ0(α) be the radially symmetric minimal solution branch for the un- perturbed problem (3.3.1) on 0 ≤ r ≤ 1. For the perturbed problem (1.1.6), λ and the outer solution u, valid for r = O(1), are expanded as u = u0 + ( −1 log δ ) u1 + · · · , λ = λ0 + ( −1 log δ ) λ1 + · · · . (3.5.6) From (1.1.6) it follows that u1 satisfies Lu1 ≡ ∆u1 + 2λ0 (1 + u0)3 u1 = λ1 (1 + u0)2 , 0 < r < 1 ; u1 = 0 , on r = 1 . (3.5.7) The matching condition to the inner solution will then lead to a Coulomb singularity for u1 as r → 0, which then completes the specification of the problem for u1. In the inner region, defined for r = O(δ), the inner variables ρ = r/δ and u = v/(− log δ) are introduced. To leading order, it follows from (1.1.6) that ∆v = 0 with v(1) = 0. Thus, v = A log ρ for some unknown constant A. In terms of outer variables this yields the matching condition u ∼ A+ ( −1 log δ ) A log r , r → 0 . The matching of the inner and outer solutions then determines A = u0(0), and that u1 49 3.6. Conclusions has the singular behaviour u1 ∼ u0(0) log r , as r → 0 . (3.5.8) The problem (3.5.7) with singular behaviour (3.5.8) determines u1 and λ1. To determine λ1(α0) from a solvability condition, Green’s identity is applied to u0α and u1 over a punctured disk ε < r < 1 in the limit ε → 0. This yields that λ1I = −2πu0(0)u0α(0), where I is given in Lemma 4.1. The result is summarized as follows: Principal Result 3.9: Consider the annulus problem (1.1.5) with δ ≪ 1, and let λ0c be the fold point location at the end of the minimal solution branch for the unperturbed problem (3.3.1). For δ ≪ 1, the fold point for (1.1.6) then satisfies λc ∼ λ0c + ( −1 log δ ) 3λ0u0(0)u0α(0) u′0α(1) +O ( −1 log δ )2 . (3.5.9) With the parameterization u0(0) = α for the unperturbed solution, the terms in (3.5.9) are calculated from the numerical solution of the unperturbed problem (3.3.1). This yields the explicit two-term expansion λc ∼ 0.789 + 1.130 ( −1 log δ ) +O ( −1 log δ )2 . (3.5.10) Owing to the logarithmic dependence on δ, the fold point location experiences a rather large perturbation even for δ rather small. This was observed in Fig. 3.3(b). 3.6 Conclusions Asymptotic expansions for the fold point location at the end of the minimal solution branch for (1.1.4) were calculated in the limiting parameter range δ ≪ 1 and δ ≫ 1 for an arbitrary domain Ω with smooth boundary. In addition, two-term asymptotic approximations for the fold point location of the fringing-field (1.1.5) and annulus prob- lems (1.1.6) were derived for small δ. The coefficients in these asymptotic expansions were evaluated numerically for both the unit slab and the unit disk. The results can be used to determine the shift in the pull-in voltage when the basic membrane problem (1.1.2) is perturbed to either (1.1.4), (1.1.5), or (1.1.6). Accurate location of the principal fold is crucial in the design of MEMS devices as such devices tend to be operated very close to this threshold value, even although exceeding it may cause the device to fail. For this reason, these techniques and results may be useful to MEMS practitioners who require accurate determination of the pull-in voltage. 50 3.6. Conclusions In the annulus problem (1.1.6), the change in topology induces a positive shift in the pull-in voltage of O(log−1 δ). This indicates that the device has a significantly higher range of operating voltages. This Chapter forms the basis of the paper [29] titled Asymptotics of some nonlinear eigenvalue problems for a MEMS capacitor: Part I: Fold point asymptotics appearing in Methods and Applications of Analysis, Vol. 15, No. 3. 51 Chapter 4 Multiple Fold Points and Singular Asymptotics The ubiquitous λ/(1+u)2 nonlinearity which appears in all MEMS equations considered here naturally motivates the question of characterizing solutions as u(x)→ −1 for some point(s) x ∈ Ω. Inspection of (1.1.2) in the eyeball norm suggests that regularity of the solution should be lost as u→ −1 and the term λ/(1 + u)2 becomes singular. Note that since physical constraints demand the deflections remain continuous. As we shall see, the manner in which this regularity is lost can generate solutions with interesting qualitative features - the most engaging of which is the infinite fold points feature first identified in [34]. In this Chapter, singular solutions of MEMS equations (1.1.4)-(1.1.6) are analyzed in the limit ε→ 0+ where ||u||∞ = 1−ε via matched asymptotic methods. The techniques developed in this Chapter allow for the resolution of the λ/(1 + u)2 nonlinearity and the construction of singular solutions in the limit ε→ 0+. In § 4.1, the infinite fold point feature of the membrane problem (1.1.3) is analyzed in the limit ε → 0+ where ||u||∞ = 1 − ε. The asymptotic analysis deployed allows for an explicit characterization of the singular solution branch, moreover, it provides an accurate asymptotic formula for the location of the fold points on the upper branch which are observed to be exponentially close to u = −1. The closely related Bratu problem is also investigated in § 4.1.3. Asymptotic predictions are observed to agree very well with numerical calculations. In § 4.2, attention is restricted to the one dimensional domain Ω = (−1, 1) and solutions to (1.1.4)-(1.1.6) are developed in the limit ε → 0+ where ε ≡ 1 + u(0). Asymptotic predictions are shown to agree well with numerical calculations. In § 4.3, the radially symmetric maximal solution branch for (1.1.4) and a pure biharmonic cousin (4.3.1) are constructed on the unit disk in the limit ε → 0+ where ε ≡ 1 + u(0). The analysis requires that the nonlinearity λ/(1 + u)2 be resolved in a local region in the vicinity of the origin. The extent of this region is determined to be O(γ), where γ satisfies the implicit relationship −γ2 log γ = ε. In § 4.3.2, the radially symmetric maximal solution branch for (1.1.6) is constructed 52 4.1. Asymptotics of the Infinite Fold Points Structure in the annulus δ ≤ |x| ≤ 1, where x ∈ R2. As the deflecting beam is pinned down at both of its end points, the location of maximum deflection occurs must be determined in the solution of the problem. A characterization of the maximal solution branch of the fringing fields problem (1.1.5) in the limit as ||u||∞ → 1 remains elusive. Finally, in § 4.4, the maximal solution branch for the pure biharmonic MEMS prob- lem (4.3.1) is constructed on an arbitrary 2D domain under the assumption the solution concentrates on a unique x0 ∈ Ω. This maximal solution is then characterized in terms of the Neumann Green’s function (4.4.2) and conditions on the concentration point x0 are established. 4.1 Asymptotics of the Infinite Fold Points Structure In this section, the bifurcation branch of radially symmetric solutions to the generalized membrane problem (1.1.3) is constructed in the limit ε→ 0+ where ||u||∞ = 1− ε. 4.1.1 Infinite Number of Fold Points For N = 1 Considered first is the case of the slab domain Ω = (−1, 1) in the parameter range α > αc, where αc = −1/2 + (3/2)3/2 is the threshold above which an infinite fold point structure exists (cf. [34]). The symmetry condition ux(0) = 0 is imposed so that attention may be restricted to the region 0 < x < 1 and so the problem uxx = λxα (1 + u)2 , 0 < x < 1; u(1) = ux(0) = 0 , (4.1.1) is considered in the limit u(0)+1 = ε→ 0+. The nonlinear eigenvalue parameter λ and the outer solution for (4.1.1), defined away from x = 0, are expanded for ε→ 0 as u = u0 + ε qu1 + · · · , λ = λ0 + εqλ1 + · · · , (4.1.2) where q > 0 is to be found. In order to match to the inner solution below, the leading- order terms u0, λ0 are taken to be the singular solution of (4.1.1) for which u0(1) = 0 and u0(0) = −1. This solution is given by u0 = −1 + xp , λ0 = p(p− 1) , p ≡ α+ 2 3 . (4.1.3) 53 4.1. Asymptotics of the Infinite Fold Points Structure By substituting (4.1.2) into (4.1.1), and equating theO(εq) terms, the following equation for u1 is determined after applying the explicit form of u0 in (4.1.3); Lu1 ≡ u1xx + 2λ0 x2 u1 = λ1x p−2 , 0 < x < 1 ; u1(1) = 0 . (4.1.4) By introducing an inner expansion, valid near x = 0, the appropriate singularity be- haviour for u1 as x→ 0 is determined. This behaviour will allow for the determination of λ1 from a solvability condition. In the inner region near x = 0, local variables y and v(y) are introduced by the definition y = x/γ , u = −1 + εv(y) . (4.1.5) Substituting (4.1.5), together with (4.1.2) for λ, into (4.1.1) gives that v′′ = γ2+α ε3 yα v2 [λ0 + ε qλ1 + · · · ] , (4.1.6) which suggests a boundary layer width of γ = ε1/p, where p is defined in (4.1.3). Expanding (4.1.6) with v = v0 + ε qv1 + · · · , (4.1.7) and equating terms at O(1) and O(εq), the following equations are obtained for v0 and v1; v′′0 = λ0y α v20 , 0 < y <∞ ; v0(0) = 1 , v′0(0) = 0 , (4.1.8a) v′′1 + 2λ0y α v30 v1 = λ1y α v20 , 0 < y <∞ ; v1(0) = v′1(0) = 0 . (4.1.8b) The leading-order matching condition is that v0 ∼ yp as y →∞. Linearizing about this far field behaviour by writing v0 = y p+w, where w≪ yp as y →∞, determines that w satisfies w′′+2λ0w/y2 = 0. This Euler type equation admits an explicit solution which determines that the far-field behaviour of the solution to (4.1.8a) is v0 ∼ yp +Ay1/2 sin (ω log y + φ) , as y →∞ , ω ≡ 1 2 √ 8λ0 − 1 , (4.1.9) where A and φ are constants depending on α, which must be computed from the nu- merical solution of (4.1.8a) with v0 ∼ yp as y → ∞. Note that 8λ0 − 1 > 0 when α > αc ≡ −1/2 + (3/2)3/2. In contrast, the far-field behaviour for (4.1.8b) is deter- mined by its particular solution. For y → ∞ we use v0 ∼ yp in (4.1.8b), to obtain 54 4.1. Asymptotics of the Infinite Fold Points Structure that v1 ∼ λ1 3λ0 yp , as y →∞ . (4.1.10) Therefore, by combining (4.1.9) and (4.1.10), the far-field behaviour of the inner expansion v ∼ v0 + εqv1 is determined. The matching condition is that this far-field behaviour as y → ∞ must agree with the near-field behaviour as x → 0 of the outer expansion in (4.1.2). By using u = −1 + εv and x = ε1/py, this matching condition yields u ∼ −1 + xp +Aε1−1/(2p)x1/2 sin ( ω log x− ω p log ε+ φ ) + εq ( λ1 3λ0 ) xp , as x→ 0 . (4.1.11) Now, comparing (4.1.11) with the outer expansion for u in (4.1.2), it is clear that u1 must solve (4.1.4), subject to the singular behaviour u1 ∼ Ax1/2 sin (ω log x+ φε) + λ1 3λ0 xp , as x→ 0 , (4.1.12) where the exponent q and the phase φε are defined to be q = 1 − 1/(2p) and φε ≡ −ωp−1 log ε+ φ. Next, we solve the problem (4.1.4) for u1, with singular behaviour (4.1.12). Since the first term in the asymptotic behaviour in (4.1.12) is a solution to the homogeneous problem in (4.1.4), while the second term is the particular solution for (4.1.4), it is convenient to decompose u1 as u1 = λ1 3λ0 xp +Ax1/2 sin (ω log x+ φε) + U1a , (4.1.13) to obtain that U1a solves LU1a = 0 , 0 < x < 1 ; U1a(1) = − λ1 3λ0 −A sinφε ; U1a = o(x1/2) , as x→ 0 . (4.1.14) To value of λ1 is determined by a solvability condition. The function Φ = x 1/2 sin(ω log x+ kπ) is a solution to LΦ = 0 with Φ(1) = 0 for any integer k, and has the asymptotic behaviour Φ = O(x1/2) and Φx = O(x−1/2) as x→ 0. By applying Lagrange’s identity to U1a and Φ over the interval 0 < σ < x < 1, and by using Φ(1) = 0, we obtain 0 = ∫ 1 σ (φLU1a − U1aLΦ) dx = −U1a(1)Φx(1)− [ΦU1ax − U1aΦx]x=σ . (4.1.15) Finally, taking the limit σ → 0 in (4.1.15) and using U1a = o(x1/2), U1ax = o(x−1/2), and Φx(1) 6= 0, as x → 0 yields that U1a(1) = 0, which determines λ1 = −3Aλ0 sinφε 55 4.1. Asymptotics of the Infinite Fold Points Structure from (4.1.14). The preceding calculation is now summarized by the following result: Principal Result 4.1: For ε ≡ u(0) + 1 → 0+ and α > αc ≡ −1/2 + (3/2)3/2 a two-term asymptotic expansion for the bifurcation curve λ versus ε of (4.1.1) is given by λ ∼ λ0 + 3Aλ0εq sin ( 3ω 2 + α log ε− φ ) + · · · , (4.1.16a) where q, ω, and λ0, are defined by q = 1− 3 2(α + 2) , ω = 1 2 √ 8λ0 − 1 , λ0 = (α− 1)(α + 2) 9 . (4.1.16b) The constants A and φ, which depend on α, are determined numerically from the solu- tion to (4.1.8a) with far-field behaviour (4.1.9). These constants are given in the first row of Table 3.1 for a few values of α. The asymptotic prediction for the locations of the infinite sequence of fold points, determined by setting dλ/dε = 0, is u(0) = −1 + εm, εm = exp [ (α+2) 3ω ( φ− (2m−1)pi2 )] , λm = λ0 + 3λ0Aε q m (−1)m, m = 1, 2, . . . (4.1.16c) These fold points are exponentially close to u(0) = −1 as ε→ 0. The analysis leading to Principal Result 4.1 is non-standard as a result of two features. First, the correction term u1 in the outer expansion for u is not independent of ε, but in fact depends on log ε. However, although u1 is weakly oscillatory in ε, it is uniformly bounded as ε→ 0. Second, the solvability condition determining λ1 pertains to a countably infinite sequence of functions Φ = x1/2 sin(ω log x + kπ) where k is an integer. 4.1.2 Infinite Number of Fold Points For N > 1 Next, a similar analysis is employed to determine the limiting form of the bifurcation diagram for radially symmetric solutions of (1.1.3) in the N -dimensional unit ball. To this end, the solution branches of urr + (N − 1) r ur = λrα (1 + u)2 , 0 < r < 1 ; u(1) = 0 , ur(0) = 0 , (4.1.17) will be constructed asymptotically in the limit u(0) + 1 = ε → 0+, where α ≥ 0 and N ≥ 2. For N = 2, the term rα represents a variable dielectric permittivity of the membrane (cf. [34], [17]). In the limit ε→ 0, (4.1.17) is a singular perturbation problem with an outer region where O(γ) < r < 1 with u = O(1), and an inner region with r = O(γ) where 56 4.1. Asymptotics of the Infinite Fold Points Structure u = O(ε). Here γ ≪ 1 is the boundary layer width to be found in terms of ε. The nonlinear eigenvalue parameter λ and the outer solution are expanded as u ∼ u0 + εqu1 + · · · , λ ∼ λ0 + εqλ1 + · · · , (4.1.18) for some q > 0 to be determined. For the leading-order problem for u0 and λ0, a singular solution of (4.1.17) is constructed for which u0(0) = −1. This singular solution is given explicitly by u0 = −1 + rp , λ0 = p2 + (N − 2)p , p ≡ (α+ 2) 3 . (4.1.19) The substitution of (4.1.18) into (4.1.17), together with using (4.1.19) for u0, shows that u1 satisfies LNu1 ≡ u′′1 + (N − 1) r u′1 + 2λ0 r2 u1 = λ1r p−2 , 0 < r < 1 ; u1(1) = 0 . (4.1.20) The required singularity behaviour for u1 as r → 0 will be determined below by matching u1 to the inner solution. In the inner region near r = 0, local variables v and ρ are introduced and their inner expansion by u = −1 + εv , v = v0 + εqv1 + · · · , ρ = r/γ , γ = ε1/p . (4.1.21) Substituting (4.1.21) into (4.1.17), reveals that v0(ρ) and v1(ρ) satisfy v′′0 + (N − 1) ρ v′0 = λ0ρ α v20 , 0 < ρ <∞ ; v0(0) = 1 , v′0(0) = 0 , (4.1.22a) v′′1 + (N − 1) ρ v′1 + 2λ0ρ α v30 v1 = λ1ρ α v20 , 0 < ρ <∞ ; v1(0) = v′1(0) = 0 . (4.1.22b) The leading-order matching condition is that v0 ∼ ρp as ρ→∞. Linearizing about this far field behaviour by writing v0 = ρ p + w, where w ≪ ρp as ρ → ∞, gives that w satisfies w′′ + (N − 1) ρ w′ + 2λ0 ρ2 w = 0 . (4.1.23) A solution to this Euler equation is w = ρµ , µ = −(N − 2) 2 ± √ (N − 2)2 − 8λ0 2 . (4.1.24) This leads to two different cases, depending on whether (N−2)2 > 8λ0 or (N−2)2 < 8λ0. 57 4.1. Asymptotics of the Infinite Fold Points Structure α = 0 α = 1 α = 2 α = 3 N A φ A φ A φ A φ 1 − − − − 1.1678 3.9932 0.8713 3.7029 2 0.4727 3.2110 0.4728 3.2110 0.4729 3.2110 0.4730 3.2109 3 0.2454 2.5050 0.2864 2.7231 0.3152 2.8351 0.3363 2.9042 4 0.1935 1.8789 0.2193 2.2932 0.2454 2.5048 0.2676 2.6347 5 0.1972 1.2755 0.1935 1.8790 0.2101 2.1886 0.2284 2.3775 6 0.2586 0.7008 0.1909 1.4743 0.1935 1.8790 0.2056 2.1262 7 0.4859 0.1945 0.2095 1.0803 0.1896 1.5746 0.1935 1.8790 Table 4.1: Numerical values of A and φ for different exponents α and dimension N computed from the far-field behaviour of the solution to (4.1.22a) with (4.1.25). Consider first the case (N − 2)2 < 8λ0 in which µ takes a complex value. As shown below, this is the case where the bifurcation diagram of λ versus ε has an infinite number of fold points. For this case, the explicit solution for w leads to the following far-field behaviour for the solution v0 of (4.1.22a): v0 = ρ p+Aρ1−N/2 sin (ωN log ρ+ φ)+o(1) , ρ→∞ ; ωN ≡ 1 2 √ 8λ0 − (N − 2)2 . (4.1.25) Here the constants A and φ, depending on N and α, must be computed numerically from the solution to (4.1.22a) with far-field behaviour (4.1.25). In Table 4.1 numerical values for these constants are given for different α and N for the parameter range where (N − 2)2 < 8λ0. The results for A and φ are also plotted in Fig. 4.1(a). In particular, for N = 2 and α = 0, A = 0.4727 , φ = 3.2110 . (4.1.26) For N = 2 and α = 0, in Fig. 4.1(b) the numerically computed far-field behaviour of v0 is plotted after subtracting off the O(ρ2/3) algebraic growth at infinity. Indeed, this far-field behaviour is oscillatory as predicted by (4.1.25). 58 4.1. Asymptotics of the Infinite Fold Points Structure 0 1 2 3 4 5 0.1 0.2 0.3 0.4 0.5 n = 2 n = 3 .. . n = 7 α A 0 1 2 3 4 5 pi 4 pi 2 3pi 4 2pi n = 2 n = 3... n = 7 α φ (a) A(α) and φ(α) for N = 2, . . . , 7. −10.0−6.0 −2.0 2.0 6.0 10.0 14.0 18.0 22.0 26.0 −0.5 0.0 0.5 1.0 log ρ v0(ρ)− ρ 2 3 (b) v0(ρ) for ρ≫ 1 Figure 4.1: Left figure: numerical results for the far-field constants A and φ in (4.1.25) for differentN and α. Right figure: Plot of v0−ρ2/3 versus log ρ as computed numerically from (4.1.22a) for N = 2 and α = 0. The far-field behaviour is oscillatory. For v1, the far-field behaviour for (4.1.22b) is determined by its particular solution. Using the far field condition v0 ∼ ρp as ρ→∞ in (4.1.22b) demonstrates that v1 ∼ λ1 3λ0 ρp , as ρ→∞ . (4.1.27) Therefore, by combining (4.1.25) and (4.1.27), the far-field behaviour of the inner expan- sion v ∼ v0 + εqv1 is obtained. The matching condition is that this far-field behaviour as ρ → ∞ must agree with the near-field behaviour as x → 0 of the outer expansion in (4.1.18). By using u = −1 + εv and r = ε1/pρ, and by choosing the exponent q in (4.1.18) appropriately, the following singularity behaviour is generated for u1 (4.1.20) subject to the singular behaviour u1 ∼ Ar1−N/2 sin (ωN log r + φε) + λ1 3λ0 rp , as r → 0 , (4.1.28) where q and φε are defined by q ≡ 1 + 3 2 ( N − 2 α+ 2 ) , φε = − 3ωN α+ 2 log ε+ φ . (4.1.29) Next, problem (4.1.20) is solve subject to the singular behaviour (4.1.28). The 59 4.1. Asymptotics of the Infinite Fold Points Structure solution of u1 is decomposed as u1 = λ1 3λ0 rp +Ar1−N/2 sin (ωN log r + φε) + U1a , (4.1.30) to obtain that U1a solves LNU1a = 0 , 0 < r < 1 ; U1a(1) = − λ1 3λ0 −A sinφε ; U1a = o(r 1−N/2) , as r → 0 . (4.1.31) Since the solution to the homogeneous problem is Φ = r1−N/2 sin (ωN log r + kπ) for any integer k ≥ 0, then Green’s second identity is readily used to obtain a solvability condition for (4.1.31). The application of this identity to U1a and Φ on the interval σ ≤ r ≤ 1 yields 0 = ∫ 1 σ rN−1 (ΦLNU1a − U1aLNΦ) dr = −rN−1 (Φ∂rU1a − U1a∂rΦ) ∣∣∣r=1 r=σ . (4.1.32) Since Φ(1) = 0, the passage to the limit σ → 0 in (4.1.32) results in U1a∂rΦ|r=1 = − lim σ→0 σN−1 (Φ∂rU1a − U1a∂rΦ) |r=σ . (4.1.33) Now since U1a = o(r 1−N/2), ∂rU1a = o(r−N/2), Φ = O(r1−N/2), and ∂rΦ = O(r−N/2) as r → 0, there is no contribution in (4.1.33) from the limit σ → 0. Consequently, U1a(1) = 0, which determines λ1 as λ1 = −3λ0A sinφε from (4.1.31). The asymptotic result is summarized as follows: Principal Result 4.2: For ε ≡ u(0)+1 → 0+ and when either N > Nc, or equivalently α > αcN , where Nc ≡ 2 + 4(α + 2) 3 + 2 √ 6 3 (α+ 2) , αcN ≡ −2 + 3(N − 2) 4 + 2 √ 6 , (4.1.34) then a two-term asymptotic expansion for the bifurcation curve λ versus ε for (4.1.17) is given by λ ∼ λ0 + 3εqAλ0 sin ( 3ωN α+ 2 log ε− φ ) , (4.1.35a) where q is defined in (4.1.29). The constants A and φ, depending on N and α, to be computed from the solution to (4.1.22a) with far-field behaviour (4.1.25), are given in Table 3.1 and Fig. 4.1(a). The asymptotic prediction for the locations of the infinite 60 4.1. Asymptotics of the Infinite Fold Points Structure sequence of fold points, determined by setting dλ/dε = 0, is u(0) = −1 + εm, εm = exp [ (α+ 2) 3ωN ( φ− (2m− 1)π 2 )] , λm = λ0 + 3λ0Aε q m (−1)m, m = 1, 2, . . . (4.1.35b) where ωN is defined in (4.1.25). The condition on N , or equivalently on α, in (4.1.34) are both necessary and suffi- cient to guarantee that 8λ0 > (N − 2)2. For α = 0, (4.1.34) yields that the dimension N satisfies 2 ≤ N ≤ 7. For any N ≥ 8, it follows from (4.1.34) that (4.1.17) has an infinite number of fold points if α is sufficiently large. Next, the case in which 8λ0 < (N − 2)2, corresponding to either N > Nc, or equivalently to α < αcN . For this case, the solution v0 to (4.1.22a) has the far-field behaviour v0 ∼ ρp +Aρµ+ , as ρ→∞ ; µ+ ≡ 1− N 2 + √ (N − 2)2 − 8λ0 2 , (4.1.36) for some constant A, which depends on N and α, that must be calculated numerically. A simple calculation shows that p > µ+. In addition, the far-field behaviour of the solution v1 to (4.1.22b) has the asymptotic behaviour in (4.1.27). Then, by using u = −1+εv and r = ε1/pρ, where v = v0+εqv1, and by choosing the exponent q in (4.1.18) appropriately, the following singularity behaviour, to be satisfied by u1, is established u1 ∼ Arµ+ + λ1 3λ0 rp , as r → 0 , (4.1.37) where q = 1− 3µ+/(2 + α). The solution of u1 admits the decomposition u1 = Arµ+ + λ1 3λ0 rp + U1a , (4.1.38) where from (4.1.20) it is determined that U1a solves LNU1a = 0 , 0 < r < 1 ; U1a(1) = − λ1 3λ0 −A ; U1a = o(rµ+) , as r → 0 . (4.1.39) The solvability condition for this problem determines λ1 as λ1 = −3λ0A. The asymp- totic result is summarized as follows: Principal Result 4.3: For ε ≡ u(0) + 1 → 0+ assume that either 2 < N < Nc, or equivalently α < αcN , where Nc and αcN are defined in (4.1.34). Then, a two-term 61 4.1. Asymptotics of the Infinite Fold Points Structure asymptotic expansion for the bifurcation curve λ versus ε for (4.1.17) is given by u(0) = −1 + ε , λ ∼ λ0 − 3εqAλ0 , (4.1.40) where q = 1−3µ+/(2 + α) and µ+ is defined in (4.1.36). The constant A, which depends on α and N , must be computed numerically from the solution to (4.1.22a) with far-field behaviour (4.1.36). λ |u(0)| 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.2 0.4 0.6 0.8 1.0 (a) N = 1, . . . , 7 and α = 0. 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 0.0 0.2 0.4 0.6 0.8 1.0 λ |u(0)| α = 0 α = 1 α = 2 (b) N = 8 and α = 0, 1, 2. Figure 4.2: Panel (a): numerical bifurcation curves |u(0)| versus λ for N = 1, . . . , 7 and α = 0. Panel (b): numerical bifurcation curves for N = 8 and α = 0, 1, 2. The results were computed numerically from (4.1.17). In Fig. 4.2 plots are shown for the bifurcation diagrams of |u(0)| versus λ, as com- puted numerically from (4.1.17) for N = 1, . . . , 7 and α = 0 (see Fig. 4.2(a)) and for N = 8 with α = 0, 1, 2 (see Fig. 4.2(b)). For representative values of N and α, in Fig. 4.3 it is observed that the asymptotic results for the bifurcation diagram as obtained from either (4.1.35) or (4.1.40) closely approximate the numerically computed bifurcation curves of (4.1.17) 62 4.1. Asymptotics of the Infinite Fold Points Structure λ |u(0)| 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.2 0.4 0.6 0.8 1.0 (a) N = 2 and α = 0, 1, 2. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.4 0.8 1.2 1.6 2.0 λ |u(0)| (b) N = 4 and α = 0. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 1.0 2.0 3.0 4.0 5.0 λ |u(0)| (c) N = 8 and α = 0. Figure 4.3: Comparison of bifurcation curves for N = 2 with α = 0, 1, 2 (Panel(a)), N = 4 and α = 0 (Panel (b)), and N = 8 with α = 0 (Panel(c)). The solid curves are the full numerical results and the dashed curves are the asymptotic results obtained from either (4.1.35) or (4.1.40). 4.1.3 Fold Points of The Bratu Problem In this section, the analysis of § 4.1.2 is extended to study a similar infinite fold points structure present in radially symmetric solutions of the Bratu Problem ∆u+ λeu = 0, x ∈ Ω; u = 0, x ∈ ∂Ω (4.1.41) where Ω is the unit ball in RN and ∆u ≡ urr + (N − 1)r−1ur for r = |x|. The analysis is performed in the limit as u(0)→∞ by setting u(0) = −2 log ε and studying the limit ε→ 0+. Solutions of (4.1.41) have been investigated thoroughly and have been shown to exhibit many curious features. One such feature, which is of main interest here, is the Infinite fold points feature exhibited when 3 ≤ N ≤ 9. An inner region is introduced in the vicinity of the origin and an outer region else- 63 4.1. Asymptotics of the Infinite Fold Points Structure where. In the outer region a solution of form u = u0 + ǫ pu1 + · · · , λ = λ0 + ǫpλ1 + · · · (4.1.42) is implemented in equation (4.1.41) where the scaling p = p(N) is to be determined later by matching with the inner problem. Inserting expansion (4.3.3) in equation (4.1.41) and collecting terms of similar order results in the following ODEs satisfied by u0 and u1; d2u0 dr2 + N − 1 r du0 dr + λ0e u0 = 0, 0 < r < 1; u0(1) = 0; (4.1.43a) d2u1 dr2 + N − 1 r du1 dr + λ0e u0u1 = −λ1eu0 , 0 < r < 1; u1(1) = 0. (4.1.43b) In equation (4.1.43a) an anzatz solution of form u0(r) = A log r is given and accepted provided A = −2 and λ0 = 2(N−2). Inputing u0 and λ0 into equation (4.1.43b) reduces the equation satisfied by u1 to d2u1 dr2 + N − 1 r du1 dr + λ0 r2 u1 = −λ1 r2 , 0 < r < 1; u1(1) = 0. (4.1.44) The required singularity behaviour for u1 as r → 0 will be determined below by matching u1 to the inner solution. In this inner region, the change of variables y = γx, u = −2 log ǫ+ v(y) (4.1.45) is implemented to transform (4.1.41) to 1 γ2 ( d2v dy2 + N − 1 y dv dy ) + λ ǫ2 ev = 0, y > 0; v(0) = 0 v′(0) = 0 . Balancing all terms with γ = ǫ results in the inner problem d2v dy2 + N − 1 y dv dy + λev = 0, y > 0; v(0) = 0 v′(0) = 0 . (4.1.46) Expanding (4.1.46) with v = v0 + ε pv1 + · · · λ = λ0 + εpλ1 + · · · 64 4.1. Asymptotics of the Infinite Fold Points Structure gives the following problems d2v0 dy2 + N − 1 y dv0 dy + λ0e v0 = 0, y > 0; v0(0) = 0 v′0(0) = 0 . (4.1.47a) d2v1 dy2 + N − 1 y dv1 dy + λ0e v0v1 = −λ1ev0 , y > 0; v1(0) = 0 v′1(0) = 0 . (4.1.47b) As y → ∞, impose behaviour v0(y) → −2 log y and look for a correction term to this leading order behaviour by writing v(y) ∼ −2 log y + w with w(y) satisfying d2w dy2 + N − 1 y dw dy + λ0 y2 w = 0. By writing w = yβ it is observed that β satisfies a quadratic equations with roots (2−N)± √ (N − 2)(N − 10) 2 . These roots are complex when 3 ≤ N ≤ 9 and it is for this range that infinite fold points are present. Thus, the behaviour of (4.1.47a) as y →∞ is summarized by v0(y) ∼ −2 log y +Ay1− N 2 sin(ωN log y + φ) + · · · where A(N) and φ(N) are determined by numerical solution of (4.1.47a) and ωN = √ (N − 2)(10 −N) 2 . (4.1.48) The far field behaviour of the v1 equation, formulated in (4.1.47b) is now developed. The equation for v1 admits the decomposition v1 = −λ1/λ0 + vH where d2vH dy2 + N − 1 y dvH dy + λ0e v0vH = 0. (4.1.49) Using the condition v0 ∼ −2 log y as y → ∞, the far field behaviour of vH is observed to be O(y1−N2 ) as y →∞. This determines the far field condition of (4.1.46) to be v ∼ −2 log y +Ay1−N2 sin(ωN log y + φ) + εp [ −λ1 λ0 +O(y1−N2 ) ] + · · · (4.1.50) Returning to an expression in outer variables via (4.1.45) gives the matching behaviour 65 4.1. Asymptotics of the Infinite Fold Points Structure with which the outer solution must agree. Therefore as r → 0 the matching requires that u ∼ −2 log r +AǫN2 −1r1−N2 sin (ωN log r + φε) + ǫp [ −λ1 λ0 +O(εN2 −1) ] + · · · (4.1.51) A comparison with equation (4.1.42) as r → 0 reveals that p = N 2 − 1, φε(N) = φ− ω(N) log ǫ. (4.1.52) where again A and φ are determined by numerical solution of (4.1.47a). Therefore, singularity condition (4.3.26) gives that u ∼ −2 log r + ε1−N2 [ −λ1 λ0 +Ar1− N 2 sin (ωN log r + φε) ] +O(εN−2) (4.1.53) which in turn furnishes the problem for u1 (4.1.44) with singularity behaviour as r → 0 to give that d2u1 dr2 + N − 1 r du1 dr + λ0e u0u1 = −λ1eu0 , 0 < r < 1; u1(1) = 0. (4.1.54a) u1 ∼ −λ1 λ0 +Ar1− N 2 sin (ωN log r + φε) + · · · r → 0 (4.1.54b) Decomposing the solution to (4.1.54) as u1 = −λ1 λ0 +Ar1− N 2 sin (ωN log r + φε) + U1H (4.1.55) it is observed that U1H satisfies d2U1H dr2 + N − 1 r dU1H dr + λ0 r2 U1H = 0, 0 < r < 1; U1H(1) = λ1 λ0 −A sinφε (4.1.56) and additionally U1H = o(r 1−N 2 ) as r → 0. Since the solution to the homogeneous problem is Φ = r1−N/2 sin (ωN log r + kπ) for any integer k ≥ 0, then Green’s second identity is readily used to obtain a solvability condition for (4.1.56). The application of this identity to U1H and Φ on the interval σ ≤ r ≤ 1 yields 0 = ∫ 1 σ rN−1 (ΦLNU1H − U1HLNΦ) dr = −rN−1 (Φ∂rU1H − U1H∂rΦ) ∣∣∣r=1 r=σ . (4.1.57) 66 4.1. Asymptotics of the Infinite Fold Points Structure Since Φ(1) = 0, the passage to the limit σ → 0 in (4.1.57) results in U1H∂rΦ|r=1 = − lim σ→0 σN−1 (Φ∂rU1H − U1H∂rΦ) |r=σ . (4.1.58) Now since U1H = o(r 1−N/2), ∂rU1H = o(r−N/2), Φ = O(r1−N/2), and ∂rΦ = O(r−N/2) as r → 0, there is no contribution in (4.1.58) from the limit σ → 0. Consequently, U1H(1) = 0, which determines λ1 as λ1 = λ0A sinφε from (4.1.56). With λ1 determined, the asymptotic bifurcation diagram of solutions is given by |u(0)| = −2 log ǫ λ = λ0 + λ0Aǫ N 2 −1 sinφε +O(εN−2) (4.1.59) with λ0 = 2(N − 2), ωN = √ (2−N)(N − 10) 2 . The fold points of the solution branch may now be located by observing the values of ǫm which satisfy φε ≡ φ− ωN log ǫm = (2m− 1)π 2 for m = 1, 2, 3, . . .. This expression can be rearranged to give ǫm = exp ( φ ωN − (2m+ 1)π 2ωN ) . The coordinates of the fold points (αm, λm) can now be given asymptotically as αm = (2m+ 1)π ωN − 2φ ωN , λm = λ0 + λ0Aǫ N 2 −1 m (−1)m (4.1.60) The constants A(N) and φ(N) are determined from the numerical solution of the initial value problem d2v dy2 + N − 1 y dv dy + 2(N − 2)ev = 0, y > 0; v(0) = 0 v′(0) = 0 . (4.1.61) Recall that the behaviour of v(y) was earlier in (4.3.25) given as v(y) ∼ −2 log y +Ay1−N2 sin(ωN log y + φ) + · · · as y → ∞. The sinusoidal term is isolated by peeling off the leading order behaviour, multiplying by y N 2 −1 and plotting the remainder against log y at which point the re- quired constants are easily noted. The following table gives approximate values of A and φ for relevant values of N : 67 4.1. Asymptotics of the Infinite Fold Points Structure N 3 4 5 6 7 8 9 A 0.990 0.605 0.493 0.489 0.581 0.850 1.716 φ 6.232 5.646 5.071 4.522 3.971 3.501 3.097 −10.0 −6.0 −2.0 2.0 6.0 10.0 14.0 18.0 22.0 −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 log y (v(y) + 2 log y)y 1 2 Figure 4.4: Numerical solution of equation (4.1.61) for N = 3. The leading order behaviour has been peeled off and a logarithmic scale is used on the x-axis to demonstrate the sinusoidal element of the solution. The following figure provides a comparison between full numerics and asymptotic pre- diction λ 0.0 1.0 2.0 3.0 4.0 |u(0)| 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 Figure 4.5: Solid line: Numerical solution of equation (4.1.41) for N = 3. Dashed line: Asymptotic solution of (4.1.41) for N = 3. The results in Figure.4.5 show that the asymptotic prediction is very good at the first fold point and indistinguishable from the full numerical solution by the second fold point. 68 4.2. Asymptotics of The Maximal Solution Branch as λ→ 0: Slab Domain 4.2 Asymptotics of The Maximal Solution Branch as λ→ 0: Slab Domain In this section, the limiting asymptotic behaviour of the maximal solution branch for the fringing field problem (1.1.5) as well as the beam problem (1.1.4) in a slab domain is constructed. 4.2.1 The Fringing Field Problem The method of matched asymptotic expansions to construct the maximal solution branch (λε, uε) of uxx = λ(1 + δu2x) (1 + u)2 , −1 < x < 1 ; u(±1) = 0 , (4.2.1) in the limit where ε ≡ u(0) + 1 → 0+. The condition that λε → 0 as ε → 0+ is made explicit by assuming that λε ∼ ν(ε)λ0 + · · · , (4.2.2) where ν(ε) ≪ 1 is to be determined. Since the solution to (4.2.1) is even about x = 0, the interval 0 < x < 1 is considered together with the condition ux(0) = 0. In the outer region, away from x = 0, the solution is expanded as uε = u0 + ν(ε)u1 + · · · , λε = ν(ε)λ0 + · · · . (4.2.3) Substituting this expansion into (4.2.1), and collecting powers of ν, gives the following sequence of problems u0xx = 0, 0 < x < 1 ; u0(1) = 0 , (4.2.4) u1xx = λ0(1 + δu 2 0x) (1 + u0)2 , 0 < x < 1 ; u1(1) = 0 . (4.2.5) By fixing u0(0) = −1, the solution to (4.2.4) and (4.2.5) is given in terms of an unknown constant a1 as u0 = −1 + x , u1 = −λ0 (1 + δ) log x+ a1(x− 1) . (4.2.6) By introducing the inner variable y = x/γ, (4.2.6) suggests that to leading order u ∼ −1 + γy + · · · as x → 0. Since u = −1 + O(ε) in the inner region, this motivates 69 4.2. Asymptotics of The Maximal Solution Branch as λ→ 0: Slab Domain the introduction of the local variables y and v defined by y = x/ε , u = −1 + εv(y) . (4.2.7) Applying variables (4.2.7) to equations (4.2.1) together with λε ∼ νλ0, gives v′′ = ε−1ν [λ0 + · · · ] v2 ( 1 + δ ( v′ )2) , (4.2.8) which indicates the correct scaling ν = ε. With ν = ε, the two-term outer approximation from (4.2.6) and (4.2.3) in terms of x = εy has the local behaviour u ∼ −1 + εy + (−ε log ε) [λ0 (1 + δ)] + ε [−λ0(1 + δ) log y − a1] + · · · . The constant term of order O(ε log ε) cannot be matched by the inner solution. There- fore, to remove this term, the outer expansion in (4.2.3) must be modified by introducing a switchback term. The modified outer expansion is uε ∼ u0 + (−ε log ε)u1/2 + εu1 + · · · , λε ∼ ελ0 + (−ε2 log ε)λ1 + · · · . (4.2.9) Upon substituting (4.2.9) into (4.2.1), we obtain that u1/2xx = 0 with u1/2(1) = 0. This gives u1/2 = a0(x− 1). The two-term outer expansion is uε ∼ −1 + x+ (−ε log ε) a0(x− 1) + ε [−λ0(1 + δ) log x+ a1(x− 1)] + · · · . (4.2.10) In terms of the inner variable, y = ε−1x, the local behaviour as x→ 0 of (4.2.10) is u ∼ −1 + εy + (−ε log ε) [−a0 + λ0(1 + δ)] + ε [−λ0(1 + δ) log y − a1] + (−ε2 log ε) ya0 + · · · . (4.2.11) To eliminate the constant term of order O(ε log ε), the necessary choice for a0 is a0 = λ0(1 + δ) . (4.2.12) Then, with u = −1 + εv, (4.2.11) yields the following required far-field behaviour for the inner solution v(y): v ∼ y − λ0(1 + δ) log y − a1 + (−ε log ε) [λ0(1 + δ)y] + · · · , y →∞ . (4.2.13) For the inner problem (4.2.8), the far-field form (4.2.13) suggests the following ex- 70 4.2. Asymptotics of The Maximal Solution Branch as λ→ 0: Slab Domain pansion for the inner solution v = v0 + (−ε log ε) v1 + · · · . (4.2.14) Then, from (4.2.14) and (4.2.13), together with (4.2.9) for λε, we obtain from (4.2.8) that v0 and v1 satisfy v′′0 = λ0 v20 ( 1 + δ(v′0) 2 ) , y > 0 ; v0(0) = 1 , v ′ 0(0) = 0 , (4.2.15a) v0 ∼ y − λ0(1 + δ) log y − a1 , as y →∞ , (4.2.15b) Lv1 ≡ v′′1 + 2λ0 v30 v1 − 2λ0δv ′ 0 v0 ( v1 v0 )′ = λ1 v20 ( 1 + δ(v′0) 2 ) , y > 0 ; v1(0) = v ′ 1(0) = 0, (4.2.15c) v1 ∼ λ0(1 + δ)y + · · · , as y →∞ . (4.2.15d) The solution to these problems determines λ0, λ1, and the unknown constant a1. The two cases δ = 0 and δ > 0 are now considered. For the case δ = 0, equation (4.2.15a) can be integrated directly after multiplying through by v′0. Upon using the far-field behaviour v ′ 0 → 1 as y →∞, the value λ0 = 1/2 is obtained and v0 satisfies v′0 = √ 1− 1 v0 , 0 < y <∞ ; v0(0) = 1 , which in turn can be integrated to yield √ v0 √ v0 − 1 + log (√ v0 + √ v0 − 1 ) = y . For v0 ≫ 1, corresponding to v0−1, this relation gives that v0+ 12 log v0+log 2−1/2 ∼ y, so that v0 ∼ y − 1 2 log y + 1 2 − log 2 , as y →∞ . Upon comparing this expression with (4.2.15b), the value a1 = log 2− 1/2 is obtained. Next, to determine λ1 when δ = 0, we first multiply the equation for v1 in (4.2.15c) by v′0. Then, by using Green’s second identity, together with Lv′0 = 0 and the boundary 71 4.2. Asymptotics of The Maximal Solution Branch as λ→ 0: Slab Domain conditions in (4.2.15) for v0 and v1 at y = 0 and as y →∞, we obtain lim R→∞ ∫ R 0 ( v′0Lv1 − v1Lv′0 ) dy = lim R→∞ ( v′0v ′ 1 − v1v′′0 ) ∣∣∣R 0 , λ1 lim R→∞ ∫ R 0 v′0 v20 dy = lim R→∞ v′0(R)v ′ 1(R) , −λ1 ∫ ∞ 0 d dy ( 1 v0 ) dy = −λ1 ( 1 v0 ) ∣∣∣∞ v0=0 = λ0 . Since v0(0) = 1 and λ0 = 1/2, the value λ1 = 1/2 is obtained. Next, consider the case δ > 0. For this case, only λ0 and a1 are readily determined analytically. To obtain λ0, let u0 = v ′ 0, and write (4.2.15a) as a first order separable ODE for u0 = u0(v0) in the form du0 dv0 = λ0 v20 ( 1 + δu20 u0 ) , 0 < v0 <∞ ; u0(1) = 0 , u0 → 1 as v0 →∞ . (4.2.16) Separating variables and integrating over 0 < v0 <∞ yields that (2δ)−1 log(1+δu20)|10 = (−λ0/v0) |∞1 , which gives λ0 = (2δ)−1 log(1+δ). Moreover, v0 = v0(y) satisfies the first- order ODE dv0 dy = δ−1/2 [ (1 + δ)1−1/v0 − 1 ]1/2 , 0 < y <∞ ; v0(0) = 1 . Integration of this ODE determines v0(y) implicitly in terms of a quadrature. By expanding this implicit integral relation as y → ∞, and then comparing the resulting expression with the required far-field behaviour in (4.2.15b), the constant a1 can be identified as a1 = ∫ ∞ 1 ([ 1 + (1 + δ) δ ( (1 + δ)−1/x − 1 )]−1/2 − 1− (1 + δ) 2δ log(1 + δ) x ) dx . (4.2.17) The determination of the second-order term λ1 from (4.2.15c) and (4.2.15d) is tedious, and is omitted. A summary of the construction of the limiting behaviour of the maximal solution branch of (4.2.1) is as follows: Principal Result 4.4: For ε ≡ u(0)+1→ 0+, the maximal solution branch of (4.2.1) has the asymptotic behaviour λε ∼ ελ0 + (−ε2 log ε)λ1 + · · · , (4.2.18a) 72 4.2. Asymptotics of The Maximal Solution Branch as λ→ 0: Slab Domain where λ0 = λ1 = 1/2 when δ = 0, and λ0 = (2δ) −1 log(1 + δ) when δ > 0. In the outer region, defined away from x = 0, a three-term expansion for uε is uε ∼ −1 + x+ (−ε log ε)λ0(1 + δ)(x− 1) + ε [−λ0(1 + δ) log x+ a1(x− 1)] , (4.2.18b) where a1 = log 2− 1/2 when δ = 0, and a1 is given in (4.2.17) when δ > 0. A favorable comparison of the full numerical solution of (4.2.1) and the asymptotic prediction λε is given in Fig. 4.6. δ = 0δ = 2δ = 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 λ |u(0)| (a) |u(0)| vs. λ for δ = 0, δ = 2, and δ = 10 δ λ0 0.0 2.0 4.0 6.0 8.0 10.0 0.5 0.4 0.3 0.2 0.1 0.0 (b) λ0 vs. δ Figure 4.6: Left figure: The full numerical solution of (4.2.1) is favorably compared with the asymptotic predictions of Principal Result 3.1 for the values δ = 0, 2, 10. Good agreement is observed when ε ≡ 1− |u(0)| is small. Right figure: Plot of λ0 vs. δ from λ0 = (2δ) −1 log(1 + δ). As a remark, the asymptotics in (4.2.18a) for the case δ = 0 can be verified by calculating the exact solution to (4.2.1). Defining u(0) = α, we multiply the equation for u in (4.2.1) by ux and integrate the resulting expression. By using ux(0) = 0, the following integral relation is obtained; λ = (1 + α) 2 [J(α)]2 , J(α) ≡ ∫ 0 α √ 1 + u u− α du . (4.2.19) Changing variables and defining α = −1 + ε, the integral J can be re-written and calculated as J = 2 ∫ 1 √ ε x2√ x2 − ε dx = √ 1− ε+ ε [log (1 +√1− ε)− log(√ε)] . (4.2.20) Substituting (4.2.20) into (4.2.19), and expanding as ε → 0, it is readily established that λ ∼ ε/2 − ε2 log(ε)/2 + · · · , in agreement with (4.2.18a) for the case δ = 0. 73 4.2. Asymptotics of The Maximal Solution Branch as λ→ 0: Slab Domain 4.2.2 The Beam Problem In this subsection, the method of matched asymptotic expansions is used to construct the maximal solution branch (λε, uε), of the pure biharmonic nonlinear eigenvalue problem − uxxxx = λ (1 + u)2 , −1 < x < 1 ; u(±1) = ux(±1) = 0 , (4.2.21) in the limit where ε ≡ u(0)+ 1→ 0+. The assumption that λε → 0 as ε→ 0+ is made, so that λε ∼ ν(ε)λ0 + · · · , (4.2.22) where ν(ε)≪ 1 is a gauge function to be determined. Since uε is even in x, the treatment of equation (4.2.21) may be restricted to the interval 0 < x < 1 in conjunction with the symmetry conditions ux(0) = uxxx(0) = 0. In the outer region for 0 < x < 1, the solution is initially expanded as uε ∼ u0 + ν(ε)u1 + · · · . (4.2.23) where u0 and u1 satisfy the following equations on 0 < x < 1: u0xxxx = 0 , 0 < x < 1 ; u0(1) = u0x(1) = 0 , (4.2.24a) u1xxxx = − λ0 (1 + u0)2 , 0 < x < 1 ; u1(1) = u1x(1) = 0 . (4.2.24b) For (4.2.24a), the point constraints u0(0) = −1 and u0x(0) = 0 are imposed in order to match to a local solution in the vicinity of x = 0. This determines u0(x) as u0(x) = −1 + 3x2 − 2x3 . (4.2.25) Since u0xxx(0) 6= 0, this leading-order outer solution does not satisfy the symmetry condition uxxx(0) = 0 indicating that an inner layer near x = 0 is required. To determine the behaviour of the O(ν) term, equation (4.2.25) is inserted to (4.2.24b), to yield for x≪ 1 that u1xxxx = − λ0 (3x2 − 2x3)2 = − λ0 9x4 ( 1− 2x 3 )−2 ∼ − λ0 9x4 ( 1 + 4x 3 + 12x2 9 + · · · ) . By integrating this limiting relation, the local behaviour u1 ∼ λ0 54 log x− 2λ0 27 x log x+ c1 + b1x+O(x2 log x) , as x→ 0 , (4.2.26) 74 4.2. Asymptotics of The Maximal Solution Branch as λ→ 0: Slab Domain is established in terms of arbitrary constants c1 and b1 which pertain to a solution of the homogeneous problem and will be specified later to determine u1 uniquely. From (4.2.23), (4.2.25), and (4.2.26), the following asymptotic behaviour for uε as x → 0 is deduced: uε ∼ −1 + 3x2 − 2x3 + ν ( λ0 54 log x− 2λ0 27 x log x+ c1 + b1x+O(x2 log x) ) + · · · . (4.2.27) By rescaling the vicinity of the origin with variable y = x/γ, the leading order behaviour from (4.2.27) is u ∼ −1 + 3γ2y2 + · · · as x→ 0. Since u = −1 +O(ε) in the inner region, this motivates the scaling of the solution in the vicinity of the origin by using local variables y and v defined by y = x/ε1/2 , u = −1 + εv(y) . (4.2.28) Balancing the cubic term −2x3 in (4.2.27) with the O(ν) term in (4.2.27) gives the scaling ν = ε3/2. Substituting (4.2.28) and λε ∼ ε3/2λ0 into (4.2.21), the problem for v(y) satisfies − v′′′′ = ε1/2 [λ0 + · · · ] v2 , 0 < y <∞ ; v(0) = 1 , v′(0) = v′′′(0) = 0 . (4.2.29) The correct expansions for the inner and outer solutions are now determined by expressing the local behaviour of the outer expansion in (4.2.27) in terms of the inner variable x = ε1/2y with ν = ε3/2, to get uε = −1 + 3εy2 + ( ε3/2 log ε ) λ0 108 + ε3/2 ( −2y3 + λ0 54 log y + c1 ) + (−ε2 log ε) λ0 27 y + ε2 [ −2λ0 27 y log y + b1y ] +O(ε5/2 log ε) . (4.2.30) The terms of order O(ε3/2 log ε) and order O(ε2 log ε) cannot be matched to the inner solution and so they must be removed by the addition of switchback terms to the outer expansion. Additionally, the O(ε) term in (4.2.30) and equation (4.2.29) indicate that v ∼ v0 + o(1), where v0 satisfies v′′′′0 = 0 , 0 < y <∞ ; v0(0) = 1 , v′0 = v′′′0 (0) = 0 ; v0 ∼ 3y2 as y →∞ , (4.2.31a) and so admits the exact solution v0 = 3y 2 + 1 . (4.2.31b) 75 4.2. Asymptotics of The Maximal Solution Branch as λ→ 0: Slab Domain The constant term in (4.2.31b) then generates the unmatched term ε in the outer region, which can only be removed by introducing a second switchback term into the outer expansion. This suggests that λε, and the outer expansion for uε, must have the form uε = u0+εu1/2+ ( ε3/2 log ε ) u3/2+ε 3/2u1+· · · , λε = ε3/2λ0+ε2λ1+· · · . (4.2.32) Upon substituting (4.2.32) into (4.2.21), and collecting similar terms in ε, u1/2 is ob- served to satisfy u1/2xxxx = 0 , 0 < x < 1 ; u1/2(0) = 1 , u1/2x(0) = b1/2 , u1/2(1) = u1/2x(1) = 0 , (4.2.33a) where b1/2 is a constant to be found. The condition u1/2(0) = 1 accounts for the constant term in v0. The solution is u1/2(x) = 1 + b1/2x+ (−3− 2b1/2)x2 + (b1/2 + 2)x3 . (4.2.33b) Similarly, u3/2(x) satisfies u3/2xxxx = 0. To eliminate the O(ε3/2 log ε) and O(ε2 log ε) terms in (4.2.30), u3/2 should satisfy u3/2xxxx = 0 together with the conditions u3/2(0) = − λ0 108 , u3/2x(0) = λ0 27 , u3/2(1) = u3/2x(1) = 0 . (4.2.34a) The solution for u3/2 is u3/2 = λ0 ( − 1 108 + x 27 − 5x 2 108 + x3 54 ) . (4.2.34b) Substituting (4.2.25), (4.2.26), (4.2.33b), (4.2.34b), for u0, u1, u1/2, and u3/2 respec- tively, into the outer expansion (4.2.32), and writing the resulting expression in terms of the inner variable x = ε1/2y generates the following behaviour for uε as x→ 0: u ∼ −1 + ε (3y2 + 1) + ε3/2(−2y3 + λ0 54 log y + b1/2y + c1 ) + ε2 ( −(3 + 2b1/2)y2 − 2λ0 27 y log y + b1y + · · · ) . (4.2.35) The local behaviour (4.2.35) suggests that the correct expansion of the inner solution is v = v0 + ε 1/2v1 + εv2 + · · · . (4.2.36) Substituting (4.2.36) and (4.2.32) for λε into (4.2.29), shows that v0 satisfies (4.2.31), 76 4.2. Asymptotics of The Maximal Solution Branch as λ→ 0: Slab Domain v1 satisfies v′′′′1 = − λ0 v20 , 0 < y <∞ ; v1(0) = v′1(0) = v′′′1 (0) = 0 , (4.2.37a) v1 ∼ −2y3 + λ0 54 log y + b1/2y + c1 , as y →∞ , (4.2.37b) while v2 satisfies v′′′′2 = 2λ0 v30 v1 − λ1 v20 , 0 < y <∞ ; v2(0) = v′2(0) = v′′′2 (0) = 0 , (4.2.38a) v2 ∼ − ( 3 + 2b1/2 ) y2 − 2λ0 27 y log y + b1y + · · · , as y →∞ . (4.2.38b) The solution to these problems fixes the values of λ0, λ1, b1/2, c1 and b1, as is now demonstrated. To calculate λ0, (4.2.37a) is integrated from 0 < y < R using v0 = (3y 2 + 1) to get lim R→∞ v′′′1 ∣∣∣R 0 = −λ0 ∫ ∞ 0 1 (3y2 + 1)2 dy = −λ0 9 ∫ ∞ 0 1 (y2 + 1/3)2 dy = −λ0 9 ( 3 √ 3π 4 ) . Using the limiting behaviour v1 ∼ −2y3 as y →∞ determines that λ0 = 48 √ 3/π. The value of b1/2 is determined by the following application of Green’s second iden- tity, lim R→∞ ∫ R 0 ( v0v ′′′′ 1 − v1v′′′′0 ) dy = lim R→∞ ( v0v ′′′ 1 − v′0v′′1 + v′′0v′1 − v′′′0 v1 ) ∣∣∣R 0 . Using v0(y) = 3y 2 + 1 together with the problem for v1, (4.2.37), we obtain that lim R→∞ ∫ R 0 v0v ′′′′ 1 dy = lim R→∞ [( 3R2 + 1 ) (−12)− 6R(−12R) + 6 (−6R2 + b1/2)+ · · · ] , −λ0 lim R→∞ ∫ R 0 1 v0 dy = −λ0 ∫ ∞ 0 1 3y2 + 1 dy = −12 + 6b1/2 . (4.2.39) Evaluating the integral and using λ0 = 48 √ 3/π results in b1/2 = −2. To determine c1 in (4.2.37), the value of v ′′ 1 (0) is calculated so that (4.2.37) may be posed as an initial value problem for v1. To determine v ′′ 1 (0), multiply (4.2.37) by v ′ 0, 77 4.2. Asymptotics of The Maximal Solution Branch as λ→ 0: Slab Domain and integrate over 0 < y < R, to get lim R→∞ ∫ R 0 v′0v ′′′′ 1 dy = lim R→∞ ( v′0v ′′′ 1 − v′′0v′′1 ) ∣∣∣R 0 + lim R→∞ ∫ R 0 v′′1v ′′′ 0 dy , −λ0 lim R→∞ ∫ R 0 v′0 v20 dy = v′′0(0)v ′′ 1 (0) + lim R→∞ [6R(−12)− 6(−12R) + · · · ] , λ0 lim R→∞ ( 1 v0 ) ∣∣∣∞ v0=1 = 6v′′1 (0) . (4.2.40) Since v0(0) = 1, this yields that v ′′ 1(0) = −λ0/6. Then, with the initial values v1(0) = v′1(0) = v ′′′ 1 (0) = 0, and v ′′ 1 (0) = −λ0/6, (4.2.37) can be integrated explicitly for v1 to obtain v1 = − λ0 12 √ 3 y3 tan−1( √ 3y)− λ0 36 y2+ λ0 108 log(1+ 3y2)− λ0 12 √ 3 y tan−1( √ 3y) . (4.2.41) Using the large argument expansion tan−1(z) ∼ π/2 − z−1 + (3z3)−1 for z → ∞ in (4.2.41), shows that v1 ∼ − λ0π 24 √ 3 y3 + λ0 54 log y − λ0π 24 √ 3 y + 2λ0 81 + λ0 log 3 108 , as y →∞ . (4.2.42) Since λ0 = 48 √ 3/π, a comparison of (4.2.42) with the required far-field behaviour for v1 in (4.2.37b), determines c1 as c1 = λ0 ( 2 81 + log 3 108 ) , λ0 = 48 √ 3 π . (4.2.43) Next, λ1 is calculated from (4.2.38) by integrating (4.2.38a) over 0 < y < ∞, and applying the condition v′′′2 → 0 as y →∞, to obtain that λ1 ∫ ∞ 0 1 v20 dy = 2λ0 ∫ ∞ 0 v1 v30 dy . Then, since v0 = 3y 2 + 1, the above expression simplifies to λ1 = 8 √ 3λ0 π ∫ ∞ 0 v1 (3y2 + 1)3 dy . (4.2.44) This integral can be directly evaluated using the known form of v1, given in equation (4.2.41). In this way, we have that λ1 ≡ 8 √ 3λ0 π [ − λ0 12 √ 3 I1 − λ0 36 I2 + λ0 108 I3 − λ0 12 √ 3 I4 ] (4.2.45) 78 4.2. Asymptotics of The Maximal Solution Branch as λ→ 0: Slab Domain where I1 = ∫ ∞ 0 y3 tan−1( √ 3y) (1 + 3y2)3 dy = −1 12 ∫ ∞ 0 [ 1 (1 + 3y2)2 ] y y2 tan−1( √ 3y) dy = 1 12 ∫ ∞ 0 [ 2y tan−1( √ 3y) (1 + 3y2)2 + √ 3y2 (1 + 3y2)3 ] dy = − 1 36 ∫ ∞ 0 [ 1 1 + 3y2 ] y tan−1( √ 3y) dy + 1 4 √ 3 ∫ ∞ 0 y2 (1 + 3y2)3 dy = 1 12 √ 3 ∫ ∞ 0 dy (1 + 3y2)2 + 1 4 √ 3 ∫ ∞ 0 y2 dy (1 + 3y2)3 = 1 12 √ 3 ( π 4 √ 3 ) + 1 4 √ 3 ( π 48 √ 3 ) = 5π 576 . I2 = ∫ ∞ 0 y2 dy (1 + 3y2)3 = π 48 √ 3 . I3 = ∫ ∞ 0 log(1 + 3y2) (1 + 3y2)3 dy = π(12 log 2− 7) 32 √ 3 . I4 = ∫ ∞ 0 y tan−1( √ 3y) (1 + 3y2)3 dy = −1 12 ∫ ∞ 0 [ 1 (1 + 3y2)2 ] y tan−1( √ 3y) dy = 1 12 ∫ ∞ 0 dy (1 + 3y2)3 = π 64 . After some simplification, the following compact expression for λ1 is obtained: λ1 = λ20 108 (3 log 2− 4) ≈ −12.454. (4.2.46) Finally, b1 is calculated from (4.2.38) by first multiplying (4.2.38a) by v0 followed by integration of the resulting expression over 0 < y < R to get ∫ R 0 v0v ′′′′ 2 dy = 2λ0 ∫ R 0 v1 v20 dy − λ1 ∫ R 0 1 v0 dy . (4.2.47) Since v1 ∼ −2y3 and v0 ∼ 3y2 as y →∞, then v1/v20 ∼ −2/(9y) as y →∞ and therefore the first integral on the right-hand side of (4.2.47) diverges as R→∞. Equation (4.2.47) must therefore be rewritten as[ v0v ′′′ 2 ∣∣∣R 0 − v′0v′′2 ∣∣∣R 0 + ∫ R 0 v′′0v ′′ 2 dy ] = 2λ0 ∫ R 0 ( v1 v20 + 2 9(y + 1) ) dy − 4λ0 9 log (R+ 1)− λ1 ∫ R 0 1 v0 dy . (4.2.48) 79 4.2. Asymptotics of The Maximal Solution Branch as λ→ 0: Slab Domain By using v0 = 3y 2 + 1 and the asymptotic behaviour of v2 as y → ∞ in (4.2.38b), the limit R→∞ may now be taken to get b1 = −λ0 27 + λ0 3 ∫ ∞ 0 ( v1 v20 + 2 9(y + 1) ) dy − 2λ0 3 ∫ ∞ 0 v1 v30 dy . (4.2.49) In the preceding calculation, the result ∫∞ 0 v −1 0 dy = π/(2 √ 3) has been used. The following is a summary of the asymptotic result: Principal Result 4.5: For ε ≡ u(0)+1→ 0+, the maximal solution branch of (4.2.21) has asymptotic behaviour λε ∼ 48 √ 3 π ( ε3/2 + 4ε2 3 √ 3π ( 3 log 2− 4)+ · · ·) , (4.2.50a) In the outer region, defined away from x = 0, a four-term expansion for uε is uε = −1 + 3x2 − 2x3 + ε ( 1− 2x+ x2)+ (ε3/2 log ε) u3/2 + ε3/2u1 + · · · . (4.2.50b) Here u3/2 is given in terms of λ0 = 48 √ 3/π by (4.2.34b), and u1 is defined uniquely in terms of c1 and b1 of (4.2.43) and (4.2.49), respectively, by the boundary value problem (4.2.24b) with singular behaviour (4.2.26). A very favorable comparison of numerical and asymptotic results for |u(0)| versus λ is given in Fig. 4.7. The two-term approximation for λ in (4.2.50a) is shown to be rather accurate even for λ not too small. 0.8 0.85 0.9 0.95 1.0 0.0 0.5 1.0 1.5 2.0 2.5 λ |u(0)| Figure 4.7: The full numerical result (solid line) for |u(0)| = 1 − ε versus λ for the biharmonic MEMS problem (4.2.21) is compared with the one-term (dotted curve) and the two-term (dashed curve) asymptotic result given in (4.2.50a). A similar analysis can be done for the more general mixed biharmonic MEMS prob- lem, written as − uxxxx + βuxx = λ̃ (1 + u)2 , |x| < 1 ; u(±1) = ux(±1) = 0 , (4.2.51a) 80 4.2. Asymptotics of The Maximal Solution Branch as λ→ 0: Slab Domain where β and λ̃ have the following definition δ = O(1) by β = δ−1 , λ = δλ̃ . (4.2.51b) For a fixed δ = O(1), the limiting behaviour of the maximal solution branch (λ̃ε, uε) is constructed in the limit uε(0) + 1 = ε→ 0+. Equation (4.2.51a) may be considered on the interval 0 ≤ x < 1 with symmetry conditions ux(0) = uxxx(0) = 0. As for the pure biharmonic problem (4.2.21), the expansion for λ̃ε and the outer expansion for uε is (see (4.2.32)), uε ∼ u0+ εu1/2+ ε3/2 log ε u3/2+ ε3/2u1+ · · · , λ̃ε ∼ ε3/2λ̃0+ ε2λ̃1+ · · · . (4.2.52) Upon substituting (4.2.52) into (4.2.51), and imposing the point constraints u0(0) = −1 and u0x(0) = 0, we obtain that u0 and u1 satisfy −u0xxxx + βu0xx = 0 , 0 < x < 1 ; u0(0) = −1 , u0x(0) = 0 u0(1) = u0x(1) = 0 , (4.2.53a) −u1xxxx + βu1xx = λ̃0 (1 + u0)2 , 0 < x < 1 ; u1(1) = 0 , u1x(1) = 0 . (4.2.53b) Moreover, the two switchback terms u1/2 and u3/2 are taken to satisfy −u1/2xxxx + βu1/2xx = 0 , 0 < x < 1 ; u1/2(0) = 1 , u1/2x(0) = b1/2 , u1/2(1) = u1/2x(1) = 0 (4.2.54a) −u3/2xxxx + βu3/2xx = 0 0 < x < 1 ; u3/2(0) = c3/2 , u3/2x(0) = b3/2 u3/2(1) = u3/2x(1) = 0 (4.2.54b) for some constants c3/2, b3/2, and b1/2, to be found. The solution to (4.2.53a) for u0 is given by u0(x) = −1 + C [ cosh( √ βx)− 1 ] +D [√ βx− sinh (√ βx )] , (4.2.55a) where C and D are constants given in terms of β by C = [√ β coth (√ β 2 ) − 2 ]−1 , D = [√ β − 2 tanh (√ β 2 )]−1 . (4.2.55b) 81 4.2. Asymptotics of The Maximal Solution Branch as λ→ 0: Slab Domain From (4.2.55), the local behaviour of u0 is determined to be u0 ∼ −1 + αx2 + γx3 + αβ 12 x4 + · · · , as x→ 0 , (4.2.56a) where α and γ are given by α ≡ 1 2 u0xx(0) = ( β 2 )[√ β coth (√ β 2 ) − 2 ]−1 , γ ≡ 1 6 u0xxx(0) = − ( β3/2 6 )[√ β − 2 tanh (√ β 2 )]−1 . (4.2.56b) As δ → ∞, corresponding to β → 0, (α, γ) → (3,−2), which agrees with the result for u0 given in (4.2.25) for the pure biharmonic case. As required for the analysis below, it is readily established from (4.2.56b) that α > 0 and γ < 0 for all δ ≥ 0. Next, from the problem (4.2.53b) for u1, the local behaviour as x→ 0 for u1 is u1 ∼ λ̃0 6α2 log x+ γλ̃0 α3 x log x+ c1 + b1x , as x→ 0 , (4.2.57) where c1 and b1, representing unknown coefficients associated with the homogeneous solution for u1, are to be determined. Next, the local behaviours (4.2.56a) and (4.2.57) for u0 and u1, respectively, are used together with the local behaviour for u1/2 and u3/2 from (4.2.54), to obtain the following near-field behaviour as x → 0 of the outer expansion in (4.2.52): u ∼ −1 + αx2 + γx3 + αβ 12 x4 + · · · ε ( 1 + b1/2x+ u1/2xx(0) x2 2 + · · · ) + ε3/2 log ε ( c3/2 + b3/2x+ · · · ) + ε3/2 ( λ̃0 6α2 log x+ γλ̃0 α3 x log x+ c1 + b1x+ · · · ) . (4.2.58) In terms in the inner variable y, defined by x = ε1/2y, (4.2.58) becomes u ∼ −1 + ε (αy2 + 1) + (ε3/2 log ε) ( c3/2 + λ̃0 12α2 ) + ε3/2 ( γy3 + b1/2y + λ̃0 6α2 log y + c1 ) + ( ε2 log ε )( b3/2y + γλ̃0 2α3 y ) + ε2 ( u1/2xx(0) y2 2 + γλ̃0 α3 y log y + αβ 12 y4 + b1y ) + · · · . (4.2.59) 82 4.2. Asymptotics of The Maximal Solution Branch as λ→ 0: Slab Domain To eliminate the O(ε3/2 log ε) and O(ε2 log ε) terms, which cannot be matched by the inner solution, the following values must be chosen for b3/2 and c3/2: c3/2 = − λ̃0 12α2 , b3/2 = − γλ̃0 2α3 . (4.2.60) With c3/2 and b3/2 given by (4.2.60), the solution u3/2 to (4.2.54b) is determined ex- plicitly. In the vicinity of the origin, local variables y = x/ε1/2 and εv(y) = 1 + u(ε1/2y) are introduced, with v(y) taking on expansion v = v0 + ε 1/2v1 + εv2 + · · · . Then, from (4.2.51) and (4.2.52) for λ̃ε, the leading-order inner solution is v0 = αy 2 + 1, and v1 satisfies v′′′′1 = − λ̃0 v20 , 0 < y <∞ ; v1(0) = v′1(0) = v′′′1 (0) = 0 , (4.2.61a) v1 ∼ γy3 + b1/2y + λ̃0 6α2 log y + c1 , as y →∞ , (4.2.61b) while v2 is the solution of v′′′′2 = 2λ̃0 v30 v1 − λ̃1 v20 + βv′′0 , 0 < y <∞ ; v2(0) = v′2(0) = v′′′2 (0) = 0 , (4.2.62a) v2 ∼ αβ 12 y4 + u1/2xx(0) y2 2 + γλ̃0 α3 y log y + b1y + · · · , as y →∞ . (4.2.62b) By repeating the analysis of the pure biharmonic case in (4.2.39)–(4.2.49), the values of λ̃0, λ̃1, b1/2, c1, and b1, can be determined from (4.2.61) and (4.2.62). In fixing λ̃1 and b1 from (4.2.62), the decomposition v2 = ṽ2 + αβy 4/12 must be made to obtain a problem for ṽ2 without the v ′′ 0 = 2α term in (4.2.62a). From the problem (4.2.61) for v1, it is shown that λ̃0 = −24γ √ α π , b1/2 = 3γ α = − √ β coth (√ β 2 ) , c1 = λ̃0 α2 ( 2 9 + logα 12 ) . (4.2.63) By using a simple scaling relation to transform (4.2.61) to (4.2.37), with solution (4.2.41), the solution to (4.2.61) for v1 is v1 = − λ̃0 12 √ α y3 tan−1( √ αy)− λ̃0 12α y2 + λ̃0 12α2 log(1 + αy2)− λ̃0 4α3/2 y tan−1( √ αy) . (4.2.64) In terms of the solution v1, the following values are determined from the problem (4.2.62) 83 4.2. Asymptotics of The Maximal Solution Branch as λ→ 0: Slab Domain for v2: λ̃1 = 8λ̃0 √ α π ∫ ∞ 0 v1 v30 dy , b1 = λ̃0γ 2α3 + λ̃0 α ∫ ∞ 0 ( v1 v20 − γ α2(y + 1) ) dy − 2λ̃0 α ∫ ∞ 0 v1 v30 dy , (4.2.65) Here v0 = αy 2+1. The integral term in the formulation of λ1 can be evaluated explicitly as was done for the pure biharmonic case (c.f. (4.2.45)-(4.2.46)) to give the compact expression λ̃1 = λ20 12α2 ( 3 log 2− 4). (4.2.66) The following summarizes the asymptotic result for (4.2.51): Principal Result 4.6: For ε ≡ u(0)+1→ 0+, the maximal solution branch of (4.2.51) has the asymptotic behaviour λ̃ε ∼ −24γ √ α π ( ε3/2 − 2γε 2 α3/2π ( 3 log 2− 4)+ · · ·) , (4.2.67a) where α and γ are defined in terms of β = δ−1 by (4.2.56b). In the outer region, defined away from x = 0, a four-term expansion for uε is uε ∼ u0 + εu1/2 + ε3/2 log ε u3/2 + ε3/2u1 + · · · . (4.2.67b) Here u0 is given explicitly in (4.2.55), while u1/2 and u3/2 are the solutions of (4.2.54) in terms of the coefficients c3/2 and b3/2, defined in (4.2.60), and b1/2 given in (4.2.63). Finally, u1 satisfies (4.2.53b), subject to the local behaviour (4.2.57), where c1 and b1 are defined in (4.2.63) and (4.2.65), respectively. We conclude this section with a few remarks. First, note that since α > 0 and γ < 0 for all δ > 0, the limiting behaviour in (4.2.67a) satisfies λ̃ε > 0, and is defined for all δ > 0. For δ → ∞, for which α → 3 and γ → −2, (4.2.67a) agrees with the pure biharmonic case result in (4.2.50a). Alternatively, for 0 < δ ≪ 1, a direct calculation of (4.2.56b) shows that α ∼ [ 2 √ δ ]−1 and γ ∼ − [6δ]−1, so that λε ≡ δλ̃ε has the small δ behaviour λε ∼ ε3/2δ−1/42 √ 2/π, which is not uniformly valid when ε3/2δ−1/4 = O(1). Finally, we remark that the two switchback terms can be written explicitly in the form u1/2(x) = w(x; 1, b1/2) and u3/2(x) = w(x; c3/2, b3/2), where w(x;w0.w1) is the solution to −wxxxx + βwxx = 0 with w(0) = w0, wx(0) = w1, w(1) = wx(1) = 0, given explicitly by w(x) = w0 + w1x+ C [ cosh (√ βx ) − 1 ] +D [√ βx− sinh (√ βx )] , (4.2.68a) 84 4.3. Asymptotics of Maximal Solution Branch as λ→ 0: Unit Disk where C and D is the unique solution to the 2× 2 linear algebraic system C [ cosh (√ β ) − 1 ] +D [√ β − sinh (√ β )] = −(w0 + w1) , (4.2.68b) C √ β sinh (√ β ) +D √ β [ 1− cosh (√ β )] = −w1 . (4.2.68c) In Fig. 4.8 shows a favorable comparison between the two-term asymptotic result (4.2.67a) and the full numerical result for λε = δλ̃ε when δ = 0.1 and δ = 1.0. 0.0 0.2 0.4 0.6 0.8 1.0 0.5 0.6 0.7 0.8 0.9 1.0 λ |u(0)| (a) δ = 0.1 0.5 0.6 0.7 0.8 0.9 1.0 0.0 1.0 2.0 3.0 4.0 5.0 λ |u(0)| (b) δ = 1.0 Figure 4.8: The full numerical result (solid line) for |u(0)| = 1 − ε versus λ for the mixed biharmonic MEMS problem (4.2.51) is compared with the one-term (dotted curve) and the two-term (dashed curve) asymptotic result given in (4.2.67a) for δ = 0.1 and δ = 1.0. 4.3 Asymptotics of Maximal Solution Branch as λ→ 0: Unit Disk 4.3.1 The Beam Problem For the unit disk in R2, the limiting behaviour of the maximal solution branch of the pure biharmonic nonlinear eigenvalue problem ∆2u = − λ (1 + u)2 , 0 < r < 1 ; u(1) = ur(1) = 0 , (4.3.1) and the mixed biharmonic problem δ∆2u−∆u = − λ (1 + u)2 , 0 < r < 1 ; u(1) = ur(1) = 0 , (4.3.2) are constructed. Here ∆u ≡ urr + r−1ur and δ > 0. In each case, as with those considered in previous sections, the definition u(0) + 1 = ε is made and a solution (λε, uε) is constructed such that λε ∼ ν(ε)λ0 as ε→ 0+, where ν(ε) is a gauge function 85 4.3. Asymptotics of Maximal Solution Branch as λ→ 0: Unit Disk to be determined and satisfies ν(ε) → 0 as ε → 0+. Since the asymptotic analyses for (4.3.1) and (4.3.2) are very similar, a full analysis is demonstrated for the pure biharmonic case, while the main results for the mixed problem are merely stated. The main challenge in constructing the asymptotic solution of (4.3.1) as u(0)→ −1+ is determining the gauge function ν(ε) and also the correct expansion of uε. This is achieved by matching an inner solution valid in a small neighbourhood of the origin to an outer expansion valid elsewhere. The initial naive expansions for λ and the outer solution are of the form uε = u0 + ν(ε)u1 + · · · , λε = ν(ε)λ0 + · · · . (4.3.3) Substituting (4.3.3) into (4.3.1), gives the following equations on 0 < r < 1 satisfied u0 and u1 ∆2u0 = 0 , u0(1) = u0r(1) = 0 ; ∆ 2u1 = − λ0 (1 + u0)2 , u1(1) = u1r(1) = 0 . (4.3.4) By imposing the point constraints u0(0) = −1 and u0r(0) = 0, the solution to (4.3.4) for u0 is u0 = −1 + r2 − 2r2 log r , (4.3.5) while u1 satisfies ∆2u1 = − λ0 r4(1− 2 log r)2 , 0 < r < 1 ; u1(1) = u ′ 1(1) = 0 . (4.3.6) Note that u0r(0) = 0, while u0rr(r) diverges as r → 0. This indicates that a boundary layer in the vicinity of r = 0 is required in order to satisfy the required symmetry condition urrr(0) = 0. To determine the behaviour of u1 as r → 0, new variables η = − log r and w(η) = u (e−η) are introduced so that η →∞ as r → 0. Using these new co-ordinates, (4.3.6), becomes w′′′′ + 4w′′′ + 4w′′ = − λ0 (1 + 2η)2 = − λ0 4η2 ( 1 + 1 2η )−2 . (4.3.7) By using (1 + h)−2 ∼ 1− 2h+ 3h2 + · · · , for h≪ 1, (4.3.7) is solved asymptotically to get w ∼ λ0 16 log η − λ0 32η − 3λ0 128η2 − 11λ0 384η3 + · · · , as η →∞ . Returning to the co-ordinates of the original problem with η = − log r, the local be- 86 4.3. Asymptotics of Maximal Solution Branch as λ→ 0: Unit Disk haviour for u1 u1 ∼ λ0 16 log(− log r)+ λ0 32 log r − 3λ0 128 log2 r + 11λ0 384 log3 r +a1+a2 log r+· · · , as r→ 0 , (4.3.8) is established where a1 and a2 are constants related to the solution of the homogeneous problem. By determining these constants below, and then by satisfying the two bound- ary conditions u1(1) = u1r(1) = 0, the solution u1 to (4.3.6) can be found uniquely. For r → 0, the two-term outer expansion for u, given by (4.3.3), has the limiting behaviour u ∼ −1 + r2 − 2r2 log r + ν ( λ0 16 log(− log r) + λ0 32 log r − 3λ0 128 log2 r + 11λ0 384 log3 r + a1 + a2 log r ) + · · · . (4.3.9) In the vicinity of the origin, local variables v and ρ are introduced for (4.3.1) with the usual definition u = −1 + ε v(ρ) , ρ = r/γ , (4.3.10) where γ ≪ 1 is the boundary layer width to be found. Upon setting r = γ ρ in (4.3.9), the leading order behaviour u = −1 + γ2 ρ2 − 2γ2 ρ2(log γ + log ρ) +O [ν log(− log γ)] , (4.3.11) is established for γ ≪ 1. The largest term of (4.3.11) must be O(ε) if the outer and inner expansions are to be successfully matched. In addition, since u0rr(0) is infinite, we expect that urr(0) = (ε/γ 2)v′′(0) is not finite as ε→ 0+. These considerations show that the boundary layer width is determined implicitly in terms of ε by − γ2 log γ = ε , and σ = − 1 log γ , (4.3.12) Using this scaling, the local behaviour (4.3.9) is written in terms of the inner variable ρ, as u ∼ −1 + 2ρ2ε+ σε(−2ρ2 log ρ+ ρ2) + ν ( −λ0 16 log σ + λ0 16 log(1− σ log ρ) −λ0 32 σ (1− σ log ρ) − 3λ0σ 2 128(1 − σ log ρ)2 − 11λ0σ 3 384(1 − σ log ρ)3 + a1 − a2 σ + a2 log ρ ) +· · · . (4.3.13) 87 4.3. Asymptotics of Maximal Solution Branch as λ→ 0: Unit Disk Expanding terms in (4.3.13) for σ ≪ 1, and collecting terms at each order, results in u ∼ −1 + ε [ 2ρ2 + σ(−2ρ2 log ρ+ ρ2) + ν ε [ − λ0 16 log σ + a1 − a2 σ + a2 log ρ+ σ ( −λ0 16 log ρ− λ0 32 ) + σ2 ( −λ0 32 log2 ρ− λ0 32 log ρ− 3λ0 128 ) + σ3 ( −λ0 48 log3 ρ− λ0 32 log2 ρ− 3λ0 64 log ρ− 11λ0 384 ) + · · · ]] . (4.3.14) To determine the scaling of ν, first substitute the local variables (4.3.10) into (4.3.1) to obtain the inner problem ∆2ρv = − σ2ν ε [λ0 + · · · ] v2 , 0 < ρ <∞ ; v(0) = 1 , v′(0) = v′′′(0) = 0 , (4.3.15) where ∆2ρ is the Biharmonic operator in terms of ρ. The largest term in (4.3.14) is O(ε), which suggests the expansion of v as v = v0+σv1+O(σ2). Therefore, the only feasible scalings for ν are ν = εσ−2 or ν = εσ−1. If ν = εσ−2, then to leading order v0 satisfies ∆2ρv0 = −λ0/v20 with v0 ∼ 2ρ2 as ρ→∞, which has no solution. Therefore, set ν = ε σ = −γ2 log γ ( −1 log γ )−1 = γ2 (log γ)2 . (4.3.16) Substituting (4.3.16) for ν into (4.3.14), and recalling that u = −1 + εv, gives the far-field behaviour that the inner solution v must satisfy as ρ→∞: v ∼ − 1 σ2 a2 − log σ σ λ0 16 + 1 σ (a1 + a2 log ρ) + ( 2ρ2 − λ0 16 log ρ− λ0 32 ) + σ ( −2ρ2 log ρ+ ρ2 − λ0 32 log2 ρ− λ0 32 log ρ− 3λ0 128 ) + σ2 ( −λ0 48 log3 ρ− λ0 32 log2 ρ− 3λ0 64 log ρ− 11λ0 384 ) + · · · . (4.3.17) The expansion of (4.3.15) with v = v0 + σv1 + O(σ2), indicates that the O(σ−2), O(σ−1), and O (σ−1 log σ), terms in (4.3.17) are too large and must be removed. This is achieved by adjusting the outer expansion by introducing switchback terms. The modi- fied outer and nonlinear eigenvalue expansions, in place of the naive original expansions (4.3.3), are 88 4.3. Asymptotics of Maximal Solution Branch as λ→ 0: Unit Disk u = u0 + ε log σ σ u1/2 + ε σ u1 + ε log σ u3/2 + ε u2 + εσ log σ u5/2 +εσ u3 + εσ 2 log σ u7/2 + εσ 2 u4 + · · · , (4.3.18a) λ = ε σ [ λ0 + σλ1 + σ 2λ2 + σ 3λ3 + · · · ] . (4.3.18b) Such a lengthy expansion is required in order to completely specify the inner solution v(ρ) up to terms of order O(σ). Upon substituting (4.3.18) into (4.3.1), the following problems are obtained on 0 < r < 1 for the switchback terms where j = 1, . . . , 4. ∆2u(2j−1)/2 = 0 ; u(2j−1)/2(0) = f(2j−1)/2 , ∂ru(2j−1)/2(0) = 0 , u(2j−1)/2(1) = ∂ru(2j−1)/2(1) = 0 , (4.3.19) These problems admit solutions u(2j−1)/2(r) = f(2j−1)/2 ( 1− r2 + 2r2 log r) , j = 1, . . . , 4 , (4.3.20) where f(2j−1)/2 for j = 1, . . . , 4 are constants to be determined. Moreover, for j = 2, 3, 4, uj(r) satisfies ∆2uj = − λj−1 (1 + u0)2 , 0 < r < 1 ; uj(1) = ∂ruj(1) = 0 , j = 2, 3, 4 . (4.3.21) The asymptotic behaviour of the solution for uj as r → 0 for j = 2, 3, 4 is (see equation (4.3.8)),   u2 u3 u4   ∼   λ1 λ2 λ3   ( log(− log r) 16 + 1 32 log r − 3 128 log2 r + 11 384 log3 r ) +   b1 c1 d1  +   b2 c2 d2   log r + · · · . (4.3.22) Here b1, b2, c1, c2, d1, and d2, are constants pertaining to the homogeneous solution. These constants are fixed in the matching process below, which then determines uj for j = 2, 3, 4 uniquely. From (4.3.18), (4.3.20), and (4.3.22), together with u = −1 + εv, the matching condition between the outer and inner solutions leads to the following 89 4.3. Asymptotics of Maximal Solution Branch as λ→ 0: Unit Disk far-field behaviour for the inner solution v as ρ→∞. v ∼ − 1 σ2 a2 + log σ σ ( f1/2 − λ0 16 ) + log σ ( f3/2 − λ1 16 ) + σ log σ ( f5/2 − λ2 16 ) + σ2 log σ ( f7/2 − λ3 16 ) + 1 σ (a1 − b2 + a2 log ρ) + ( b1 − λ0 32 + c2 + 2ρ 2 + ( b2 − λ0 16 ) log ρ ) + σ ( −2ρ2 log ρ+ ρ2 − λ0 32 log2 ρ+ ( c2 − λ0 32 − λ1 16 ) log ρ+ c1 − d2 − λ1 32 − 3λ0 128 ) + σ2 ( −λ0 48 log3 ρ− ( λ1 32 + λ0 32 ) log2 ρ+ ( d2 − λ2 16 − λ1 32 − 3λ0 64 ) log ρ + d1 − e2 − λ2 32 − 3λ1 128 − 11λ0 384 ) +O(σ3) (4.3.23) The constant e2 in the order O(σ2) term above arises from the homogeneous component to the solution u5 of the εσ 3u5 term in the outer expansion, not explicitly written in (4.3.18a). Since the expansion of the inner solution v(ρ) is v = v0 + σv1 + σ 2v2 + · · · , (4.3.24) the constant terms in (4.3.23), which are larger than O(1), and the O(σp log σ) terms in (4.3.23), must all be eliminated. This lead us to choose values a1 = b2 , a2 = 0 ; f(2j−1)/2 = λj−1 16 , j = 1, . . . , 4 , (4.3.25) so that (4.3.23) becomes v ∼ [ b1 − λ0 32 + c2 + 2ρ 2 + ( b2 − λ0 16 ) log ρ ] + σ [ −2ρ2 log ρ+ ρ2 − λ0 32 log2 ρ+ ( c2 − λ0 32 − λ1 16 ) log ρ+ c1 − d2 − λ1 32 − 3λ0 128 ] + σ2 [ −λ0 48 log3 ρ− ( λ1 32 + λ0 32 ) log2 ρ+ ( d2 − λ2 16 − λ1 32 − 3λ0 64 ) log ρ + d1 − e2 − λ2 32 − 3λ1 128 − 11λ0 384 ] +O(σ3) (4.3.26) From (4.3.24), (4.3.15), and (4.3.26), the leading order solution v0 satisfies ∆2ρv0 = 0 , ρ > 0 ; v0(0) = 1 , v ′ 0(0) = v ′′′ 0 (0) = 0 ; v0 ∼ 2ρ2 , as ρ→∞ . (4.3.27) 90 4.3. Asymptotics of Maximal Solution Branch as λ→ 0: Unit Disk The solution is v0 = 2ρ 2 + 1. The matching condition from the first line in (4.3.26) determines the constants b1 and b2 to be b2 = λ0 16 , b1 = 1 + λ0 32 + c2 . (4.3.28) From the O(σ) terms in (4.3.15), (4.3.24), and (4.3.26), the problem for v1 satisfies ∆2ρv1 = − λ0 v20 , ρ > 0 ; v1(0) = v ′ 1(0) = v ′′′ 1 (0) = 0 , (4.3.29a) with the far-field behaviour v1 ∼ −2ρ2 log ρ+ ρ2 − λ0 32 log2 ρ+ χ1 log ρ+ χ2 , as ρ→∞ . (4.3.29b) Here the constants χ1 and χ2 are defined by χ1 ≡ c2 − λ0 32 − λ1 16 , χ2 ≡ c1 − d2 − λ1 32 − 3λ0 128 . (4.3.29c) From the O(σ2) terms in (4.3.15), (4.3.24), and (4.3.26), the problem for v2 satisfies ∆2ρv2 = − λ1 v20 + 2λ0 v30 v1 , ρ > 0 ; v2(0) = v ′ 2(0) = v ′′′ 2 (0) = 0 , (4.3.30a) with the following far-field behaviour as ρ→∞: v2 ∼ −λ0 48 log3 ρ− ( λ1 32 + λ0 32 ) log2 ρ+ ( d2 − λ2 16 − λ1 32 − 3λ0 64 ) log ρ + d1 − e2 − λ2 32 − 3λ1 128 − 11λ0 384 + · · · . (4.3.30b) The problem (4.3.29) determines the constants λ0, χ1, and χ2. Thus, (4.3.29c) fixes c2 in terms of λ0 and λ1, and (4.3.28) determines b1. However, (4.3.29c) only provides one of the two required equations to determine c1. As shown below, the additional equation is provided by the problem (4.3.30) for v2. To determine λ0, the divergence theorem is applied to (4.3.29a) to get lim R→∞ [∫ R 0 ( ∆2ρv1 + λ0 (1 + 2ρ2)2 ) ρ dρ ] = lim R→∞ [ ρ d dρ (∆ρv1) ∣∣ ρ=R + λ0 4 ] = 0 . (4.3.31) The leading-order term in the far-field behaviour (4.3.29b) yields that ∆ρ v1 ∼ −8 log ρ as ρ→∞ and so (4.3.31) determines that λ0 = 32 and consequently a1 = 2, b1 = b2 = 2, from (4.3.25) and (4.3.28), respectively. Note that λ0 is determined solely by the leading- 91 4.3. Asymptotics of Maximal Solution Branch as λ→ 0: Unit Disk order behaviour v1 ∼ −2ρ2 log ρ, while the correction term ρ2 in (4.3.29b) specifies v1 uniquely, and allows for the determination of χ1 and χ2 in (4.3.29b) as demonstrated below To determine χ1 and χ2, equation (4.3.29a) is directly integrated to obtain ∆ρ v1 = −4 log(1 + 2ρ2) + C , (4.3.32) with the value C = 4(log 2 − 1) obtained by using the far-field behaviour (4.3.29b) in (4.3.32). Integrating (4.3.32) again with v′1(0) = 0, the following initial value problem for v1(y) is determined v1ρ = −2ρ log(ρ2 + 1/2) − ρ−1 log(1 + 2ρ2) , v1(0) = 0 . (4.3.33) A further integration of (4.3.33) yields v1 = ρ 2 − 1 2 log 2− ( ρ2 + 1 2 ) log ( ρ2 + 1 2 ) − ∫ ρ 0 log(1 + 2y2) y dy . (4.3.34) In order to identify the constants χ1 and χ2 in (4.3.29b), the asymptotic expansion of (4.3.34) as ρ→∞ must be calculated. To do so, the divergent integral in (4.3.34) must first be manipulated by re-writing it as I ≡ ∫ ρ 0 log(1 + 2y2) y dy = 1 2 ∫ 2ρ2 0 log(1 + x) x dx = 1 2 [∫ 1 0 log(1 + x) x dx+ ∫ 2ρ2 1 log(1 + x) x dx ] = π2 24 + 1 2 ∫ 2ρ2 1 log(1 + x) x dx = π2 24 + 1 2 [∫ 2ρ2 1 ( log(1 + x) x − log x x ) dx+ ∫ 2ρ2 1 log x x dx ] , = π2 24 + 1 4 [ log ( 2ρ2 )]2 + 1 2 ∫ 2ρ2 1 log(1 + 1/x) x dx , where the identity ∫ 1 0 x −1 log(1 + x) dx = π2/12 has been used. Therefore, (4.3.34) becomes v1 = −(ρ2 + 1/2) log(ρ2 + 1/2) + ρ2 − 1 2 log 2− π 2 24 − log2 ρ − log 2 log ρ− 1 4 log2 2− 1 2 ∫ 2ρ2 1 log(1 + 1/x) x dx , (4.3.35) 92 4.3. Asymptotics of Maximal Solution Branch as λ→ 0: Unit Disk where the integral term is finite as ρ→∞. Now taking the limit as ρ→∞, v1 ∼ −2ρ2 log ρ+ ρ2 − log2 ρ− (1 + log 2) log ρ − ( 1 2 + 1 2 log 2 + 1 4 log2 2 + π2 24 + 1 2 ∫ ∞ 1 log(1 + 1/x) x dx ) + o(1) , where ∫∞ 1 x −1 log(1 + x−1) dx = ∫ 1 0 u −1 log(1 + u) du = π2/12. Upon comparing this asymptotic result with (4.3.29b), the constants χ1 and χ2 are identified as χ1 = −1− log 2 , χ2 = −1 2 − 1 2 log 2− π 2 12 − 1 4 log2 2 . (4.3.36) Therefore, from (4.3.29c) and (4.3.28), the constants b1 and c2 have values c2 = λ0 32 + λ1 16 − 1− log 2 , b1 = 1 16 (λ0 + λ1)− log 2 . (4.3.37) The value of λ1 is determined by the next order problem (4.3.30) for v2 as well as additional relations for the unknown constants d1, d2, and e2. The value of λ1 is obtained by multiplying (4.3.30a) by ρ, integrating the resulting expression over 0 < ρ < R, and then using the divergence theorem. In passing to the limit R → ∞, note that since v2 = o(ρ 2 log ρ) as ρ→∞ from (4.3.30b), there is no contribution from the flux of ∆ρv2 across the big circle ρ = R. In this way, λ1 is represented by the integral λ1 = 8λ0 ∫ ∞ 0 v1 v30 ρ dρ , (4.3.38) where v0 = 2ρ 2+1 and v1 is given in (4.3.35). This expression can be evaluated explicitly with integration by parts and application of equation (4.3.33). The calculation proceeds as follows: λ1 = 8λ0 ∫ ∞ 0 ρ (1 + 2ρ2)3 v1 dρ = −λ0 ∫ ∞ 0 [ 1 (1 + 2ρ2)2 ] ρ v1 dρ = λ0 ∫ ∞ 0 [ 1 (1 + 2ρ2)2 ] v1ρ dρ = 2λ0 log 2 ∫ ∞ 0 ρ (1 + 2ρ2)2 dρ− λ0 ∫ ∞ 0 log(1 + 2ρ2) ρ(1 + 2ρ2) dρ = λ0 2 log 2− λ0 2 ∫ ∞ 0 log(1 + u) u(1 + u) du = λ0 2 ( log 2− π 2 6 ) ≈ −15.229 In the final step, the result ∫∞ 0 log(1 + u)/(u + u 2)du = π2/6 has been used. In terms of the known values of χ1, λ0, and λ1, the constant c2 is fixed by (4.3.29c). However, the expression for χ2 in (4.3.29c) gives only one equation for the two further unknowns c1 and d2. The solution v2 of (4.3.30) is uniquely defined, and so provides 93 4.3. Asymptotics of Maximal Solution Branch as λ→ 0: Unit Disk the far-field behaviour v2 = −λ0 48 log3 ρ− ( λ1 32 + λ0 32 ) log2 ρ+ χ3 log ρ+ χ4 + o(1) , as ρ→∞ , for some constants χ3 and χ4 determined in terms of the solution. A comparison of this behaviour with (4.3.30b) provides two equations χ3 = d2 − λ2 16 − λ1 32 − 3λ0 64 , χ4 = d1 − e2 − λ2 32 − 3λ1 128 − 11λ0 384 . Therefore, d2 is fixed in terms of λ2, which then determines c1 from (4.3.29c) for χ2. The equation for χ4 gives one equation for d1 and e2 in terms of λ2. This process continues to higher order to determine a further equation for d1 and e2. The preceding calculation is summarized as follows: Principal Result 4.7: In the limit ε ≡ u(0) + 1 → 0+, the limiting asymptotic be- haviour of the maximal radially symmetric solution branch of (4.3.1), away from a boundary layer region near r = 0, is given by u = u0+ ε σ log σ u1/2+ ε σ u1+ε log σ u3/2+εu2+O(εσ log σ), λ = ε σ λ0+ελ1+O(εσ) , (4.3.39a) where σ = −1/ log γ and the boundary layer width γ is determined in terms of ε by −γ2 log γ = ε. In (4.3.39a), u0 = −1 + r2 − 2r2 log r , u1/2 = − λ0 16 u0 , u3/2 = − λ1 16 u0 , (4.3.39b) while u1 and u2 are the unique solutions of ∆2u1 = − λ0 (1 + u0)2 , 0 < r < 1 ; u1(1) = u1r(1) = 0 , (4.3.39c) u1 = λ0 16 log(− log r) + λ0 16 +O(log−1 r) , r → 0 , (4.3.39d) ∆2u2 = − λ1 (1 + u0)2 , 0 < r < 1 ; u2(1) = u2r(1) = 0 , (4.3.39e) u2 = λ1 16 log(− log r) + 1 16 (λ0 + λ1)− log 2 + λ0 16 log r +O(log−1 r) , r → 0 . (4.3.39f) Finally, λ0 and λ1 are given by λ0 = 32 , λ1 = λ0 2 ( log 2− π 2 6 ) (4.3.39g) A very similar asymptotic analysis can be performed to determine the limiting behaviour 94 4.3. Asymptotics of Maximal Solution Branch as λ→ 0: Unit Disk of the maximal solution branch for the mixed biharmonic problem (4.3.2). The analysis follows very closely that of § 4.4 and so only the key results are highlighted. The main difference, as compared to the pure biharmonic case, is that now the leading-order outer solution u0 satisfies ∆2u0 − 1 δ ∆u0 = 0 , 0 < r < 1 ; u0(1) = u0r(1) = 0 , (4.3.40a) subject to the local behaviour u0 = −1 + αr2 log r + ϕr2 + o(r2) , as r→ 0 , (4.3.40b) for some α and ϕ, which are functions of δ. The general solution of (4.3.40a) is u0 = A+ B log r + CK0(ηr) +DI0(ηr) , η ≡ 1/ √ δ , (4.3.41) whereK0(z) and I0(z) are the usual modified Bessel functions. By satisfying the bound- ary conditions in (4.3.40a), together with imposing the local behaviour (4.3.40b) via the point constraints u0(0) = −1 and u0r(0) = 0, the arbitrary constants are A = [I0(η) (1 + ηK ′0(η)) − ηI ′0(η)K0(η)] G(η) , B = C = ηI ′0(η)G(η) , D = − [1 + ηK ′0(η)] G(η) , (4.3.42a) where G(η) is defined by G(η) ≡ [ηI ′0(η) (K0(η) + log (η/2) + γe) + (1 + ηK ′0(η)) (1− I0(η))]−1 . (4.3.42b) Here γe ∼ 0.5772 is Euler’s constant. Equations (4.3.40b) and (4.3.42), allow an explicit calculation of the functions α(η) and ϕ(η) in (4.3.40b); α = − ( η3 4 ) I ′0(η)G(η) , ϕ = − η2 4 [ ηI ′0(η) (log (η/2) + γe − 1) + 1 + ηK ′0(η) ]G(η) . (4.3.43) In deriving (4.3.43) and (4.3.42), the well-known small argument expansions for K0(z) and I0(z), K0(z) ∼ − [log (z/2) + γe] I0(z) + z2/4 , I0(z) ∼ 1 + z2/4 , as z → 0 . (4.3.44) have been used. This explicit solution for α and ϕ has two key properties. First, the limit δ →∞, or equivalently η → 0, corresponds to the pure biharmonic case, i.e. (α,ϕ) → (−2, 1) as η → 0, in agreement with the pure biharmonic case result (4.3.5). Our second remark 95 4.3. Asymptotics of Maximal Solution Branch as λ→ 0: Unit Disk pertains to the sign of α(η). The asymptotic analysis leading to Principal Result 4.3 below requires that α < 0 for all η > 0. This is readily verified numerically from the explicit formula for α(η) given in (4.3.43) (see Fig. 4.9(b) below). The solution is expanded, as in the pure biharmonic case, with u = u0 + ε σ log σ u1/2 + ε σ u1 + ε log σ u3/2 + εu2 +O(εσ log σ), λ = δ [ ε σ λ0 + ελ1 +O(εσ) ] , (4.3.45) where as before the fractional terms u1/2, u3/2, . . . satisfy the homogeneous problem with point constraint at r = 0 chosen to eliminate terms of order O(εσk log σ), k = −1, 0, 1, 2, . . .. The terms u1, u2, . . . satisfy ∆2uj − η2∆uj = −λj−1/(1 + u0)2 for j = 1, 2, 3, . . . and exhibit a singularity behaviour which is deduced by the methods leading to (4.3.22). After some algebra, it is established that ( see § 4.4 for more detail ) uj ∼ λj−1 4α2 [ log(− log r)− Γ1 log r + Γ2 log2 r − Γ3 log3 r ] + bj log r + cj + · · · r → 0 Γ1 = − ( 1 + ϕ α ) , Γ2 = − [ 3 4 + ϕ α + ϕ2 2α2 ] , Γ3 = − [ 1 + 3ϕ 2α + ϕ2 α2 + ϕ3 3α3 ] (4.3.46) where the terms bj log r and cj are arbitrary solutions of the homogeneous problem. In a region of width O(γ), where −γ2 log γ = ε in the vicinity of the origin, local coordinates u = −1 + εv(ρ), r = γρ are used to establish the following problem for v(ρ): δ∆2v − εσ∆v = −σ 2λ εv2 , ρ > 0; v(0) = 1, vρ(0) = vρρρ(0) = 0 (4.3.47a) v ∼ −αρ2 + ( b2 − λ0 4α2 ) log ρ+ c2 − b3 + λ0Γ1 4α2 + σ [ αρ2 log ρ+ ϕρ2 − λ0 8α2 log2 ρ+ ( b3 + λ0Γ1 4α2 − λ1 4α2 ) log ρ+ c3 − b4 + λ0Γ2 4α2 + λ1Γ1 4α2 ] + σ2 [ − λ0 12α2 log3 ρ+ ( λ0Γ1 4α2 − λ1 8α2 ) log2 ρ+ ( b4 + λ0Γ2 2α2 + λ1Γ1 4α2 − λ2 4α2 ) log ρ +c4 − b5 − λ0Γ3 4α2 + λ1Γ2 4α2 + λ2Γ1 4α2 ] +O(σ3) (4.3.47b) 96 4.3. Asymptotics of Maximal Solution Branch as λ→ 0: Unit Disk The form of far field behaviour (4.3.47b) suggests the expansion v = v0 + σv1 + σ 2v2 +O(σ3), λ = δ [ ε σ λ0 + ελ1 +O(εσ) ] . (4.3.48) which at leading order yields the solution v0 = 1− αρ2 and additionally b2 = λ0 4α2 , c2 − b3 + λ0Γ1 4α2 = 1. (4.3.49) The higher order problems which fix the corrections λ0 and λ1 are ∆2v1 = −λ0 v20 , x ∈ Ω; v1(0) = v1ρ(0) = v1ρρρ(0) = 0 (4.3.50a) v1 ∼ αρ2 log ρ+ ϕρ2 − λ0 8α2 log2 ρ+ ( b3 + λ0Γ1 4α2 − λ1 4α2 ) log ρ + c3 − b4 + λ0Γ2 4α2 + λ1Γ1 4α2 + · · · ρ→∞ (4.3.50b) and ∆2v2 = −λ1 v20 + 2λ0 v30 v1, ρ > 0; v2(0) = v2ρ(0) = v2ρρρ(0) = 0 (4.3.51a) v2 ∼ − λ0 12α2 log3 ρ+ ( λ0Γ1 4α2 − λ1 8α2 ) log2 ρ+ ( b4 + λ0Γ2 2α2 + λ1Γ1 4α2 − λ2 4α2 ) log ρ + c4 − b5 − λ0Γ3 4α2 + λ1Γ2 4α2 + λ2Γ1 4α2 + · · · ρ→∞ (4.3.51b) Equations (4.3.51) can be directly integrated to obtain that λ0 = 8α 2 and that v1ρ = αρ log(ρ 2 − 1/α) − log(1− αρ 2) ρ + ρ(α+ 2ϕ), ρ > 0; v1(0) = 0 (4.3.52) The final step in the construction of the two term asymptotic solution is a lengthy but straightforward calculation of λ1. Integrating (4.3.51a) by parts and applying (4.3.52) reveals that λ1 = −λ0 2 [ π2 6 − log(−α) + ( 1 + 2ϕ α )] . (4.3.53) Again, more detail on the preceding calculation is provided in § 4.4 which outlines the construction of the maximal solution branch on a general 2D domain. The main result characterizing the limiting form for the bifurcation diagram of (4.3.2) is as follows. Principal Result 4.8: In the limit ε ≡ u(0) + 1 → 0+, the limiting asymptotic be- haviour of the maximal radially symmetric solution branch of (4.3.2), away from a 97 4.3. Asymptotics of Maximal Solution Branch as λ→ 0: Unit Disk boundary layer region near r = 0, is given by u = u0 + ε σ log σ u1/2 + ε σ u1 + ε log σ u3/2 + εu2 +O(εσ log σ), λ = δ [ ε σ λ0 + ελ1 +O(εσ) ] , (4.3.54a) where σ = −1/ log γ and the boundary layer width γ is determined in terms of ε by −γ2 log γ = ε. In (4.3.54a), u0 = A+ B log r + CK0(ηr) +DI0(ηr) , u1/2 = − λ0 4α2 u0 , u3/2 = − λ1 4α2 u0 , (4.3.54b) where A, B, C, and D are defined in (4.3.42). Moreover, u1 and u2 are the unique solutions of ∆2u1 − η2∆u1 = − λ0 (1 + u0)2 , 0 < r < 1 ; u1(1) = u1r(1) = 0 , (4.3.54c) u1 ∼ λ0 4α2 log(− log r) + λ0 4α2 +O(log−1 r) , as r→ 0 , (4.3.54d) ∆2u2 − η2∆u2 = − λ1 (1 + u0)2 , 0 < r < 1 ; u2(1) = u2r(1) = 0 , (4.3.54e) u2 ∼ λ1 4α2 log(− log r) + λ0 4α2 log r + λ0 2α2 ( 1 + ϕ α ) + λ1 4α2 − log(−α) +O(log−1 r) , (4.3.54f) where η ≡ δ−1/2. Finally, λ0 and λ1 are given by λ0 = 8α 2 , λ1 = −λ0 2 [ π2 6 − log(−α) + ( 1 + 2ϕ α )] . (4.3.54g) Since α < 0 for all η > 0, the formulae in Principal Result 4.3 are well-defined. In Fig. 4.9, a favourable comparison of asymptotic predictions and numerical results for the limiting behaviour of the maximal solution branch is displayed. 98 4.3. Asymptotics of Maximal Solution Branch as λ→ 0: Unit Disk δ = 0.1 δ = 0.5 δ = 0.05 0.95 0.96 0.97 0.98 0.99 1.0 0.0 0.1 0.2 0.3 0.4 0.5 λ |u(0)| (a) Asymptotic predictions (dashed) and full numerical (solid) bifurcation curves for several δ. ϕ(δ) α(δ) 1.0 2.0 3.0 4.0 5.0 1.0 0.0 −1.0 −2.0 −3.0 −4.0 δ (b) α(δ) and ϕ(δ) Figure 4.9: In panel (a), the predictions of Principal Result 4.4 (dashed) are compared with full numerical solutions (solid) for several values of δ and good agreement is observed when |u(0)| is close to 1. In panel (b), α(δ) and ϕ(δ) from (4.3.43) are graphed over a range of δ. In agreement with the pure bi-harmonic case, it is observed that (α,ϕ)→ (−2, 1) as δ →∞. 4.3.2 The Annulus Problem In this subsection, the limiting behaviour of the maximal solution branch is constructed for the annulus problem ∆u = λ (1 + u)2 , δ ≤ r ≤ 1 ; u(δ) = u(1) = 0 , (4.3.55) with 0 < δ < 1 and δ = O(1). A solution of (4.3.55) for which λ→ 0 as u(rε)+1 ≡ ε→ 0 is sought, where rε is a free-boundary point to be determined. This problem, which is analyzed by formal asymptotic methods, is related to the problem studied rigorously in [65]. In the outer region, defined away from rε, we try an expansion for the outer solution u and for λ in the form u = u0 + νu1 + · · · , λ = ν (λ0 + µλ1 + · · · ) , (4.3.56) where ν ≪ 1 and µ ≪ 1 are gauge functions to be determined. As shown below, this expansion must be adjusted by inserting a certain switchback term in the outer expansion. From (4.3.56) and (4.3.55), we obtain that u0 and u1 satisfy ∆u0 = 0 , δ < r < rε , rε < r < 1 ; u0(δ) = u0(1) = 0 , (4.3.57a) ∆u1 = λ0 (1 + u0)2 , δ < r < rε , rε < r < 1 ; u1(δ) = u1(1) = 0 . (4.3.57b) 99 4.3. Asymptotics of Maximal Solution Branch as λ→ 0: Unit Disk Upon imposing the point constraint that u0(rε) = −1, the solution of (4.3.57a) gives leading order behaviour u0 =   −log(r/δ)/ log(rε/δ) , δ < r < rε −log r/ log rε , rε < r < 1 , (4.3.58) which is not differentiable at rε. Then, since u0 ∼ −1 + u0r ( r±ε ) (r − rε) as r → r±ε , (4.3.57b) yields ∆u1 = λ0(r − rε)−2/ [ u0r ( r±ε )]2 as r → rε. Therefore, u1 must have the following local behaviour as r→ r±ε : u1 ∼ − λ0[ u0r ( r±ε )]2 log |r − rε|+ a1 + o(1) , as r→ r±ε , (4.3.59) where a1 is a constant associated with the homogeneous solution to (4.3.57b). From (4.3.56), (4.3.58), and (4.3.59), the following limiting behaviour is obtained from the outer expansion u ∼ −1 + u0r(r±ε )(r − rε) + u0rr(r ± ε ) 2 (r − rε)2 + ν [ − λ0[ u0r ( r±ε )]2 log |r − rε|+ a1 + o(1) ] + · · · , as r → r±ε . (4.3.60) In the vicinity of rε, an inner solution is introduced with local variables v and ρ by u = −1 + εv(ρ) , ρ = (r − rε)/γ , where γ ≪ 1 is the internal layer width to be determined. The leading-order term in the local behaviour (4.3.60) of the outer expansion gives u ∼ −1 + γu0r(r±ε )ρ, which must match with the inner expansion u = −1 + εv. The boundary layer width γ = ε is chosen so to leading order the inner solution must have far field behaviour v ∼ u0r(r±ε )ρ as ρ → ±∞. With λ ∼ ν [λ0 + µλ1 + · · · ] and γ = ε, the following inner problem for v(ρ) is obtained from (4.3.55); v′′ + εv′ rε + ερ = ν ε [λ0 + µλ1 + · · · ] v2 , (4.3.61) which suggests the scaling ν = ε. The scale of µ relative to ε is at this stage unknown. Therefore, to leading order, set rε ∼ r0 + o(1) and v ∼ v0 to obtain that v′′0 = λ0 v20 , −∞ < ρ <∞ ; v0(0) = 1 , v ′ 0(0) = 0 ; v0 ∼ u0r(r±0 )ρ , as ρ→ ±∞ . (4.3.62) 100 4.3. Asymptotics of Maximal Solution Branch as λ→ 0: Unit Disk The condition v0(0) = 1 and v ′ 0(0) = 0, with v ′′ 0 (0) > 0, is a necessary and sufficient condition for u to have its minimum value of −1+ε at rε = r0+o(1). From (4.3.62), we conclude that v0 is even in ρ, and consequently r0 satisfies u0r(r + 0 ) = −u0r(r−0 ). From (4.3.58), this equation for r0 reduces to log r0 = −log(r0/δ), which yields r0 = √ δ. To determine λ0, multiply equation (4.3.62) for v0 by v ′ 0, and then integrate from 0 < ρ <∞ to get 1 2 [ v′0(∞) ]2 = λ0 ∫ ∞ 0 v′0 v20 dy = λ0 ∫ ∞ 1 dv0 v20 = λ0 ( − 1 v0 ) ∣∣∣∞ 1 = λ0 . (4.3.63) Then, by using v′0(∞) = u0r(r+0 ) = [r0 log r0]−2 and r0 = √ δ, we conclude from (4.3.63) that λ0 = 2 [ δ(log δ)2 ]−1 . To construct a higher-order expansion for λ, the free boundary location rε, and the inner and outer expansions for (4.3.55), further terms in the far-field behaviour of v0 as ρ → ±∞ must first be calculated. To do so, equation (4.3.62) is integrated to obtain an implicitly-defined exact solution for v0, given by √ v0(v0 − 1) + log (√ v0 + √ v0 − 1 ) = √ 2λ0ρ , for ρ ≥ 0 . (4.3.64) Since √ 2λ0 = u0r(r + 0 ) = −u0r(r−0 ), and v0(ρ) = v0(−ρ), the far-field far-field behaviour for v0 as ρ→ ±∞, obtained from (4.3.64), is v0 ∼ ±u0(r+0 )ρ− λ0[ u0r(r + 0 ) ]2 log ρ+χ , as ρ→ ±∞ ; χ ≡ 12 − log 2− 14 log(2λ0) . (4.3.65) By substituting r − rε = ερ into the local behaviour of the outer expansion (4.3.60), the condition u ∼ −1− ε log ε λ0[ u0r ( r±ε )]2 + ε [ u0r(r ± ε )ρ− λ0[ u0r ( r±ε )]2 log ρ+ a1 ] + ε2 2 u0rr(r ± ε )ρ 2 + · · · . as r → r±ε . (4.3.66) is established. The constant O(ε log ε) term in (4.3.66) cannot be accounted for by the inner expansion and so the outer expansion for u0 in (4.3.56) must be modified by inserting a switchback term. The modified outer expansion, in place of (4.3.56), is u = u0 + (−ε log ε)u1/2 + εu1 + · · · ; λ = ε (λ0 + µλ1 + · · · ) . (4.3.67) 101 4.3. Asymptotics of Maximal Solution Branch as λ→ 0: Unit Disk Upon substituting (4.3.67) into (4.3.55), the equation for u1/2(r) satisfies ∆u1/2 = 0 , δ < r < rε , rε < r < 1 ; u1/2(δ) = u1/2(1) = 0 . (4.3.68) By enforcing the point constraint u1/2(rε) = −λ0/ [ u0r(r ± ε ) ]2 , the constant term of order O(ε log ε) is eliminated in (4.3.66). In this way, the solution to (4.3.68) can be written in terms of u0 as u1/2(r) = λ0[ u0r(r ± ε ) ]2u0(r) . (4.3.69) In addition, a higher-order expansion for the free-boundary point is rε = r0 + σr1 + · · · , (4.3.70) where r0 = √ δ and the gauge function σ ≪ 1 is to be found. To determine the matching condition between the inner expansion u = −1 + ε(v0 + µv1 + · · · ) and the modified outer expansion (4.3.67), we add the local behaviour of (−ε log ε) u1/2 as r → rε to (4.3.66) and use (4.3.70) for rε and use the far-field behaviour (4.3.65) for v0. The matching condition, written in terms of the inner variable ρ, is −1 + ε [ u0r(r ± 0 )ρ− λ0[ u0r(r ± 0 ) ]2 log ρ+ a1 ] + εσr1u0rr(r ± 0 )ρ+ (−ε2 log ε)u1/2r(r±0 )ρ ∼ −1 + ε [ u0r(r ± 0 )ρ− λ0[ u0r(r ± 0 ) ]2 log ρ+ χ ] + εµv1 + · · · . (4.3.71) This matching condition yields that µ = −ε log ε , σ = −ε log ε , a1 = χ , (4.3.72) where χ is defined in (4.3.65). Since µ = −ε log ε ≫ O(ε), we conclude from (4.3.61) and (4.3.71) that the inner correction v1 satisfies Lv1 ≡ v′′1 + 2λ0 v30 v1 = λ1 v20 , −∞ < ρ <∞ , (4.3.73a) v1(0) = 0 ; v1 ∼ [ u0rr(r ± 0 )r1 + u1/2r(r ± 0 ) ] ρ , as ρ→ ±∞ . (4.3.73b) Since u0rr(r ± 0 ) = ∓u0r(r+0 )/r0 from (4.3.57a), and u1/2r(r±0 ) = ±λ0/ [ u0r(r + 0 ) ] , as ob- 102 4.3. Asymptotics of Maximal Solution Branch as λ→ 0: Unit Disk tained from the explicit solution (4.3.69), the far-field condition (4.3.73b) becomes v1 ∼ ±Aρ , as ρ±∞ ; A ≡ −r1 r0 u0r(r + 0 ) + λ0 u0r(r + 0 ) . (4.3.73c) Since v0 is an even function, the general solution to (4.3.73a) must be the sum of an even and odd function. The condition v1(0) = 0 enforces that v1 is odd, and consequently A = 0 from (4.3.73c). Since λ0 = [ u0r(r + 0 ) ]2 /2, the condition that A = 0 determines r1 as r1 = r0/2. Finally, the value of λ1 is determined from a solvability condition. Multiplying (4.3.73a) by v′0, integrating, and using Lv′0 = 0, gives∫ ∞ 0 v′0Lv1 dρ = λ1 ∫ ∞ 0 v′0 v20 dρ = Av′0(∞) . However, since A = 0, it is clear that λ1 = 0. In summary, Principal Result 4.9: In the limit ε ≡ u(rε) + 1 → 0+ for some free-boundary point rε, the limiting asymptotic behaviour of the maximal radially symmetric solution branch of (4.3.55), away from the internal layer region near r = rε, is u ∼ u0(r) + (−ε log ε) λ0 [u0r(r0)] 2u0(r) + εu1(r) + · · · . (4.3.74a) Here, u0 is given in (4.3.58) and u1 satisfies (4.3.57b) subject to u1 ∼ − λ0 [u0r (r0)] 2 log |r − r0|+ χ+ o(1) , as r→ r0 , (4.3.74b) where χ is defined in (4.3.65). Finally, λ and the free-boundary point rε are given for ε→ 0 by λ ∼ 2ε δ [log δ]2 + o(ε2 log ε) , rε ∼ √ δ [ 1 + (−ε log ε) 2 + · · · ] . (4.3.74c) In Fig. 4.10, a comparison of the asymptotic prediction for λ given in (4.3.74c) with the corresponding full numerical result computed from (4.3.55) is provided. Notice that, owing to the small error term in (4.3.74c) for λ, the asymptotic result for λ accurately predicts the full numerical result for λ even when ε is not too small. 103 4.4. Concentration Phenomena General Domains 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 1.0 λ ||u||∞ Figure 4.10: Comparison of numerical solution for ||u||∞ versus λ (heavy solid curve) computed from the annulus problem (1.1.6) in δ < r < 1 for δ = 0.1 with the limiting asymptotic approximation (4.3.74c) (dashed curve) valid for λ→ 0. 4.4 Concentration Phenomena General Domains In this section, the analysis of (4.3.1) is extended to analyze MEMS concentration for a general 2D domain. In the case of the unit disc, the symmetry of the domain led us to the assumption that the largest deflection should occur at the origin. However, for more general geometries Ω ⊂ R2, it is not clear how to predict at which point the maximum deflection of the device is obtained. To investigate this problem, solutions to ∆2u = − λ (1 + u)2 , x ∈ Ω; x ∈ ∂Ω; ||u||∞ = 1− ε (4.4.1) are constructed in the limit as ε → 0. An important tool in the analysis, is the Bihar- monic Neumann’s Green Function G(x, ξ) and its Regular Part R(x; ξ), with definition ∆2G = δ(x− ξ), x ∈ Ω; G = ∂nG = 0, x ∈ ∂Ω (4.4.2a) G(x, ξ) = 1 8π |x− ξ|2 log |x− ξ|+R(x; ξ) (4.4.2b) for ξ ∈ Ω. We suppose that the solution of (4.4.1) concentrates at a single point x0 ∈ Ω in the sense that u(x0) = −1 + ε. Additionally, it is supposed that the solution is radially symmetric in the vicinity of this concentration points. To this end, motivated by the expansion of (4.3.18), equation (4.4.1) is expanded with u = u0 + ε σ ∞∑ j=1 σj−1[uj + (− log σ)u(2j−1)/2] +O(εµ) λ = ε σ ∞∑ i=0 σjλj +O(εµ) (4.4.3) 104 4.4. Concentration Phenomena General Domains where µ≪ σk for any k ∈ N. To leading order, we have the problem ∆2u0 = 0, x ∈ Ω; u0 = ∂nu0 = 0 x ∈ ∂Ω; (4.4.4a) u0(x0) = −1, ∇xu0(x0) = 0; (4.4.4b) In terms of the Neumann’s Green Function (4.4.2), the solution of (4.4.4) may be rep- resented as u0(x;x0) = − G(x;x0) R(x0;x0) (4.4.5) The condition that ∇xu0(x0) = 0 requires ∇xR(x0;x0) = 0 and the condition that u(x) ≥ −1 requires R(x0;x0) > 0. These two conditions may be viewed as necessary conditions for solutions of (4.4.2) to concentrate at x0. Definition: Single Concentration Point. If the maximal solution branch of (4.4.1) concentrates at some unique x0 ∈ Ω, then R(x0;x0) > 0, ∇xR(x0;x0) = 0, (4.4.6) where R(x;x0) is the Regular Part of the Green’s Function defined in (4.4.2). By expanding (4.4.5) as x→ x0, the following singularity behaviour is established ∆2u0 = 0, x ∈ Ω; u0 = ∂nu0 = 0 x ∈ ∂Ω; (4.4.7a) u ∼ −1+αr2 log r + r2[β + ac cos 2θ + as sin 2θ] +O(r3), r = |x− x0| → 0 (4.4.7b) where x− x0 = r(cos θ, sin θ) and α = −1 8πR(x0;x0) , β = −1 4R(x0;x0) [ ∂2R ∂x21 + ∂2R ∂x22 ] x=x0 as = −1 2R(x0;x0) [ ∂2R ∂x1∂x2 ] x=x0 , ac = −1 4R(x0;x0) [ ∂2R ∂x21 − ∂ 2R ∂x22 ] x=x0 (4.4.7c) This completes the specification of the leading order solution. The key qualitative features of this solution are its r2 log r singularity as r → 0 which permits the point constraint u(x0) = −1 and the constraint α < 0. Proceeding to the correction terms from (4.4.3), we have that ∆2uj = −λj−1 (1 + u0)2 , x ∈ Ω; uj = ∂nuj = 0, x ∈ ∂Ω (4.4.8) for j = 1, 2, 3, . . . and with behaviour at the origin as x → x0 to be determined in the 105 4.4. Concentration Phenomena General Domains following calculation. As r = |x− x0| → 0 in (4.4.8), local behaviour (4.4.7b) is applied so that ∆2uj ∼ −λj−1 α2r4 log2 r [ 1 + β̄ α log r ]−2 , β̄ = β + as sin 2θ + ac cos 2θ. (4.4.9) To establish an asymptotic solution to (4.4.9) as r → 0, it is convenient to utilize the variable η = − log r and seek a solution to v(η, θ) = u(e−η, θ) as η → ∞. This transformation reduces (4.4.9) to vηηηη + 4vηηη + 4vηη + 4vθθ + 4vθθη + 2vθθηη + vθθθθ = −λj−1 α2η2 [ 1 + 2β̄ αη + 3β̄2 α2η2 + 4β̄3 α3η3 + · · · ] (4.4.10) A solution to (4.4.10) is developed which is accurate to O(η−3). By noting that β̄2 = β2 + a2c + a 2 s 2 + 2β(ac cos 2θ + as sin 2θ) + acas sin 4θ + a2c − a2s 2 cos 4θ β̄3 = [ β3 + 3β 2 (a2c + a 2 s) ] + 3∑ n=1 (ān cosnθ + b̄n sinnθ.) (4.4.11) the following asymptotic solution is developed as η →∞, v(η, θ) ∼ λj−1 4α2 [ log η + Γ1 η + Γ2 η2 + Γ3 η3 − [ ac 4α cos 2θ + as 4α sin 2θ ] 1 η2 + + 1 2α 1 η3 [ β α + 1 4 ] (ac cos 2θ + as sin 2θ) +O(η−4) ] (4.4.12) where the constants Γ1,Γ2 and Γ3 have values Γ1 = − ( 1 + β α ) , Γ2 = − [ 3 4 + β α + β2 2α2 + a2c + a 2 s 4α2 ] , Γ3 = − [ 1 + 3β 2α + β2 α2 + β3 3α3 + ( 1 + β α ) a2c + a 2 s 2α2 ] . (4.4.13) Lengthy expressions for the constants ān, b̄n for n = 1, 2, 3 can be obtained but these terms do not play a role in capturing behaviour to O(η−3) and so they are omitted. By returning to the variable r = e−η, singularity behaviour (4.4.12) furnishes problems 106 4.4. Concentration Phenomena General Domains (4.4.8) for j = 1, 2, 3, . . . to give; ∆2uj = −λj−1 (1 + u0)2 , x ∈ Ω; uj = ∂nuj = 0, x ∈ ∂Ω (4.4.14a) uj ∼ λj−1 4α2 [ log(− log r)− Γ1 log r + Γ2 log2 r − Γ3 log3 r − [ ac 4α cos 2θ + as 4α sin 2θ ] 1 log2 r − 1 2α 1 log3 r [ β α + 1 4 ] (ac cos 2θ + as sin 2θ) +O(log−4 r) ] + bj log r + cj + dj cos 2θ + ej sin 2θ + · · · j = 1, 2, 3. (4.4.14b) Note that the terms bj log r, cj , dj sin 2θ, ej cos 2θ relate to an arbitrary solution of the homogeneous problem. The logarithmic switchback terms, u(2j−1)/2 for j = 1, 2, 3, . . . are defined as follows ∆2u(2j−1)/2 = 0, x ∈ Ω; u(2j−1)/2 = ∂nu(2j−1)/2 = 0, x ∈ ∂Ω (4.4.15a) u(2j−1)/2(x0) = f(2j−1)/2, ∇xu(2j−1)/2(x0) = 0; (4.4.15b) u(2j−1)/2 = −f(2j−1)/2 u0; u(2j−1)/2 ∼ f(2j−1)/2 +O(r2 log r), r→ 0 (4.4.15c) Here, the f(2j−1)/2 for j = 1, 2, 3, . . . are constants whose value will be chosen to remove troublesome large terms in the inner expansion. By employing arguments similar to those used in § 4.3.1, the boundary layer width in the vicinity of the concentration point is found to be of scale O(γ) where γ2 = εσ, σ = −1 log γ . (4.4.16a) Using this information, and local co-ordinates u = −1 + εv, x− x0 = ερ(cos θ, sin θ) (4.4.16b) the nature of the expansion (4.4.3) in the vicinity of the concentration point x0 is explored. After some algebra, and retaining only terms up toO(σ2), the outer expansion 107 4.4. Concentration Phenomena General Domains (4.4.3) is expressed in terms of variables (4.4.16a) to yield that u ∼ −1 + ε [−αρ2 + σρ2(β + ac cos 2θ + as sin 2θ) + 1 4α2 [ λ0 σ + λ1 + λ2σ + · · · ] [ log(1− σ log ρ) + σΓ1 1− σ log ρ + σ2Γ2 (1− σ log ρ)2 − σ 3Γ3 (1− σ log ρ)3 − 1 4α σ2 (1− σ log ρ)2 (ac cos 2θ + as sin 2θ) + 1 2α ( β α + 1 4 ) σ3 (1− σ log ρ)3 (ac cos 2θ + as sin 2θ) ] + σ−2b1 + 3∑ j=0 σj−1[bj log ρ+ cj − bj+1 + dj cos 2θ + ej sin 2θ] − log σ 3∑ j=0 σj−1 [ λj 4α2 + f(2j−1)/2 ] Expanding for small σ and retaining terms up to O(σ2) the following expansion for v(ρ, θ) is obtained v ∼ σ−2b1 + σ−1(c1 − b2 + b1 log ρ+ d1 cos 2θ + e1 sin 2θ) + ( b2 − λ0 4α2 ) log ρ − αρ2 + c2 − b3 + λ0Γ1 4α2 + d2 cos 2θ + e2 sin 2θ − log σ 3∑ j=0 σj−1 [ λj 4α2 + f(2j−1)/2 ] + σ [ αρ2 log ρ+ ρ2(β + ac cos 2θ + as sin 2θ)− λ0 8α2 log2 ρ+ ( b3 + λ0Γ1 4α2 − λ1 4α2 ) log ρ + ( d3 − λ0ac 16α3 ) cos 2θ + ( e3 − λ0as 16α3 ) sin 2θ + c3 − b4 + λ0Γ2 4α2 + λ1Γ1 4α2 ] + σ2 [ − λ0 12α2 log3 ρ+ ( λ0Γ1 4α2 − λ1 8α2 ) log2 ρ+ ( b4 + λ0Γ2 2α2 + λ1Γ1 4α2 − λ2 4α2 ) log ρ − λ0 8α2 (ac cos 2θ + as sin 2θ) log ρ+ ( d4 − λ1ac 16α3 + λ0ac 8α3 ( β α + 1 4 )) cos 2θ + ( e4 − λ1as 16α3 + λ0as 8α3 ( β α + 1 4 )) sin 2θ ] +O(σ3) (4.4.17) The condition v(0, θ) = 1 suggests the largest term in expansion (4.4.17) should be O(1) 108 4.4. Concentration Phenomena General Domains which asserts the following values for certain constants b1 = e1 = e2 = d1 = d2 = 0; c1 = b2; f(2j−1)/2 = − λj 4α2 j = 1, 2, 3. (4.4.18) Now, applying the transformation (4.4.16) to the main equation (4.4.1), the following equation is obtained for v(ρ, θ), ∆2v = −σ 2λ εv2 , ρ > 0; v(0) = 1 vρ(0) = vρρρ(0) = 0 (4.4.19a) which is furnished with far field behaviour v ∼− αρ2 + ( b2 − λ0 4α2 ) log ρ+ c2 − b3 + λ0Γ1 4α2 + σ [ αρ2 log ρ+ ρ2(β + ac cos 2θ + as sin 2θ)− λ0 8α2 log2 ρ+ ( b3 + λ0Γ1 4α2 − λ1 4α2 ) log ρ + ( d3 − λ0ac 16α3 ) cos 2θ + ( e3 − λ0as 16α3 ) sin 2θ + c3 − b4 + λ0Γ2 4α2 + λ1Γ1 4α2 ] + σ2 [ − λ0 12α2 log3 ρ+ ( λ0Γ1 4α2 − λ1 8α2 ) log2 ρ+ ( b4 + λ0Γ2 2α2 + λ1Γ1 4α2 − λ2 4α2 ) log ρ − λ0 8α2 (ac cos 2θ + as sin 2θ) log ρ+ ( d4 − λ1ac 16α3 + λ0ac 8α3 ( β α + 1 4 )) cos 2θ + ( e4 − λ1as 16α3 + λ0as 8α3 ( β α + 1 4 )) sin 2θ + c4 − b5 − λ0Γ3 4α2 + λ1Γ2 4α2 + λ2Γ1 4α2 ] +O(σ3) (4.4.19b) The form of the far field condition (4.4.19b) suggests that the solution of (4.4.19a) should admit the expansion v = v0 + σv1 + σ 2v2 +O(σ3), λ = ε σ [ λ0 + σλ1 + σ 2λ2 +O(σ3) ] At leading order v0 = v0(ρ) satisfies ∆2v0 = 0, ρ > 0; v0(0) = 1 v0ρ(0) = v0ρρρ(0) = 0 (4.4.20a) v0 ∼ −αρ2 + ( b2 − λ0 4α2 ) log ρ+ c2 − b3 + λ0Γ1 4α2 (4.4.20b) 109 4.4. Concentration Phenomena General Domains and admits the solution v0(ρ) = 1− αρ2 which in turn specifies b2 = λ0 4α2 , c2 = 1 + b3 − λ0Γ1 4α2 . (4.4.21) Equating terms at O(σ) gives that v1(ρ, θ) satisfies ∆2v1 = −λ0 v20 , x ∈ Ω; v1(0, θ) = v1ρ(0, θ) = v1ρρρ(0, θ) = 0 (4.4.22a) with far field behaviour v1 ∼ αρ2 log ρ+ ρ2(β + ac cos 2θ + as sin 2θ)− λ0 8α2 log2 ρ+ ( b3 + λ0Γ1 4α2 − λ1 4α2 ) log ρ + ( d3 − λ0ac 16α3 ) cos 2θ + ( e3 − λ0as 16α3 ) sin 2θ + c3 − b4 + λ0Γ2 4α2 + λ1Γ1 4α2 (4.4.22b) as ρ→∞. Consider the decomposition v1(ρ, θ) = V(ρ, θ) + v̄(ρ) where ∆2V = 0, ρ > 0; V(0, θ) = Vρ(0, θ) = Vρρρ(0, θ) = 0 (4.4.23a) with far field behaviour V ∼ ρ2(ac cos 2θ + as sin 2θ) + ( d3 − λ0ac 16α3 ) cos 2θ + ( e3 − λ0as 16α3 ) sin 2θ (4.4.23b) as ρ → ∞. The solution to problem (4.4.23) is V = ρ2(ac cos 2θ + as sin 2θ) which in turn gives that d3 = λ0ac 16α3 , e3 = λ0as 16α3 (4.4.24) The problem for v̄(ρ) is now ∆2v̄ = −λ0 v20 , ρ > 0; v̄(0, θ) = v̄ρ(0, θ) = v̄ρρρ(0, θ) = 0 (4.4.25a) with far field behaviour v̄ ∼ αρ2 log ρ+ βρ2 − λ0 8α2 log2 ρ+ ( b3 + λ0Γ1 4α2 − λ1 4α2 ) log ρ+ c3 − b4 + λ0Γ2 4α2 + λ1Γ1 4α2 (4.4.25b) as ρ → ∞. By recalling that v0 = 1 − αρ2, equation (4.4.25a) is directly integrated to show that λ0 = 8α 2. (4.4.26) 110 4.4. Concentration Phenomena General Domains More information on the solution of (4.4.25) can be extracted by direct integration. Indeed, some simple algebraic manipulations show that ∆v̄ = 2α log(1− αρ2) +C (4.4.27) where the constant C can be seen from (4.4.25b) to take the value C = 4(α + β) − 2α log(−α). Integrating once more yields that v̄ρ = αρ log(ρ 2 − 1/α) − log(1− αρ 2) ρ + ρ(α+ 2β), ρ > 0; v̄(0) = 0 (4.4.28) and one final additional integration gives v̄ = α 2 ( ρ2 − 1 α ) log(ρ2 − 1/α) + βρ2 − 1 2 log(−α)− ∫ ρ 0 log(1− αx2) x dx. (4.4.29) Using manipulations very similar to those leading to (4.3.35), we have that ∫ ρ 0 log(1− αx2) x dx = π2 24 + 1 4 log2(−αρ2) + 1 2 ∫ −αρ2 1 log(1 + 1/x) x dx (4.4.30) and so the final specification of v̄(ρ) is v̄(ρ) = α 2 ( ρ2 − 1 α ) log(ρ2 − 1/α) + βρ2 − log2 ρ− log(−α) log ρ − [ π2 24 + 1 4 log2(−α) + 1 2 ∫ −αρ2 1 log(1 + 1/x) x dx ] (4.4.31) where the integral term on the right hand side of (4.4.31) is finite as ρ → ∞. In fact, (4.4.31) has far field behaviour v̄ ∼ αρ2 log ρ+ βρ2 − log2 ρ− (1 + log(−α)) log ρ− 1 2 [ π2 6 + 1 2 log2(−α) + log(−α) + 1 ] (4.4.32) as ρ → ∞ where the identity ∫∞1 x−1 log(1 + 1/x) dx = π2/12 has been used. A comparison of (4.4.32) with far field behaviour (4.4.25b) indicates that b3 = λ1 4α2 − λ0Γ1 4α2 − 1− log(−α), c3 = b4 − λ0Γ2 4α2 − λ1Γ1 4α2 − 1 2 [ π2 6 + 1 2 log2(−α) + log(−α) + 1 ] . (4.4.33) 111 4.4. Concentration Phenomena General Domains This concludes the analysis of terms at O(σ). At O(σ2), the relevant equations are ∆2v2 = −λ1 v20 + 2λ0 v30 v1, ρ > 0; v2(0) = v2ρ(0) = v2ρρρ(0) = 0 (4.4.34a) furnished with the far field condition v2 ∼ − λ0 12α2 log3 ρ+ ( λ0Γ1 4α2 − λ1 8α2 ) log2 ρ+ ( b4 + λ0Γ2 2α2 + λ1Γ1 4α2 − λ2 4α2 ) log ρ − λ0 8α2 (ac cos 2θ + as sin 2θ) log ρ+ ( d4 − λ1ac 16α3 + λ0ac 8α3 ( β α + 1 4 )) cos 2θ + ( e4 − λ1as 16α3 + λ0as 8α3 ( β α + 1 4 )) sin 2θ + c4 − b5 − λ0Γ3 4α2 + λ1Γ2 4α2 + λ2Γ1 4α2 + · · · (4.4.34b) as ρ → ∞. To fix λ1, (4.4.34a) is integrated directly over 0 < ρ < ∞. The far field behaviour gives no contribution to the flux of ∆v2, therefore λ1 ∫ 2pi 0 ∫ ∞ 0 1 v20 ρdρdθ = λ0 ∫ 2pi 0 ∫ ∞ 0 2v1 v30 ρdρdθ. Integration over the angular component makes no contribution to the integrals and so we have that λ1 = −4αλ0 ∫ ∞ 0 v̄ v30 ρdρ (4.4.35) where v̄(ρ) is the radially symmetric component of v1(ρ, θ), defined in (4.4.29). Recalling that v0 = 1− αρ2 and integrating (4.4.35) by parts yields that λ1 = −λ0 ∫ ∞ 0 v̄ [ 1 (1− αρ2)2 ] ρ dρ = λ0 ∫ ∞ 0 v̄ρ [ 1 (1− αρ2)2 ] dρ = λ0 ∫ ∞ 0 [ αρ log(ρ2 − 1/α) − log(1− αρ 2) ρ + ρ(α+ 2β) ]( 1 1− αρ2 )2 dρ = λ0[α+ 2β − α log(−α)] ∫ ∞ 0 ρdρ (1− αρ2)2 − λ0 ∫ ∞ 0 log(1− αρ2) ρ(1− αρ2) dρ = −λ0 2 [ 1 + 2β α + log(−α) ] − λ0 2 ∫ ∞ 0 log(1 + x) x(1 + x) dx = −λ0 2 [ π2 6 − log(−α) + ( 1 + 2β α )] . In the preceding calculation, the identity ∫∞ 0 log(1 + x)/(x + x 2)dx = π2/6 has been used. This completes a two-term asymptotic construction of the maximal solution to 112 4.4. Concentration Phenomena General Domains (4.4.1) and a general geometry in two dimensions. The following is a summary: Principal Result 4.10: In the limit as ε ≡ u(x0) + 1→ 0+, the limiting behaviour of the maximal solution branch of (4.4.1) has asymptotic formulation u = u0 − ε log σ σ u1/2 + ε σ u1 − ε log σu3/2 + εu2 +O(εσ log σ) λ = ε σ λ0 + ελ1 +O(εσ) (4.4.36a) where σ = −1/ log γ and the boundary layer width γ is determined in terms of ε by −γ2 log γ = ε. The point x0 ∈ Ω is assumed to be the unique point satisfying ∇R(x0;x0) = 0, R(x0;x0) > 0 (4.4.36b) where R(x;x0) is the Regular part of the Green’s function, defined in (4.4.2). In terms of this Green’s function, u0 = − G(x;x0) R(x0;x0) , u1/2 = λ0 4α2 u0, u1/2 = λ1 4α2 u0 (4.4.36c) where u0 ∼ −1 + αr2 log r + r2(β + ac cos 2θ + as sin 2θ) + · · · , (4.4.36d) as x−x0 = r(cos θ, sin θ)→ 0 and (α, β, ac, as) are given in (4.4.7c) in terms of R(x;x0). Additionally, u1 and u2 satisfy ∆u1 = − λ0 (1 + u0)2 , x ∈ Ω; u1 = ∂nu1 = 0, x ∈ ∂Ω u1 ∼ λ0 4α2 log(− log r) + λ0 4α2 +O(log−1 r), r → 0 (4.4.36e) ∆u2 = − λ1 (1 + u0)2 , x ∈ Ω; u2 = ∂nu2 = 0, x ∈ ∂Ω u2 ∼ λ1 4α2 log(− log r) + λ0 4α2 log r + λ0 2α2 ( 1 + β α ) + λ1 4α2 − log(−α) +O(log−1 r) (4.4.36f) Finally, λ0 and λ1 are given by λ0 = 8α 2, λ1 = −λ0 2 [ π2 6 − log(−α) + ( 1 + 2β α )] (4.4.36g) By assumption α < 0 and so all formulae in Principal Result 4.10 are well defined. In this calculation of the maximal solution branch, it has been assumed that the 113 4.5. Conclusions limiting solution concentrates at a unique x0 ∈ Ω satisfying condition (4.4.6). While this assumption seems reasonable for a large class of two dimensional geometries, we speculate that for certain domains, for example dumbbell shaped domains, the limiting singular solution may concentrate at multiple points. The multiplicity of such concen- tration points satisfying (4.4.6) can be determined for particular domains by a numerical investigation of the Green’s function (4.4.2). 4.5 Conclusions In this Chapter the analysis of several MEMS models has been conducted in the singular limit ||u||∞ → 1. In each case, the definition ||u||∞ = 1− ε has been made and matched asymptotic expansions have been used to resolve the λ/(1 + u)2 nonlinearity near the point of maximum deflection in the lim ε → 0+. In each of the models considered, the singular solution can be constructed and an asymptotic form of the limiting branch determined. In § 4.1, the infinite fold point structure of the membrane problem (1.1.3), originally observed by [34], is analyzed. Conditions are established on N and α under which the bifurcation diagram of (1.1.3) undergoes an infinite number of folds and an asymptotic representation of this curve is obtained. The explicit nature of our method allows for the location of each fold point to be accurately determined in terms of ε = 1+ u(0). It is observed that fold points are located exponentially close to u = −1. In the case of equations (1.1.4)-(1.1.6), it is observed numerically that the infinite fold point structure is destroyed and what remains is a maximal solution branch with limiting behaviour λ→ 0 as ||u||∞ → 1. This limiting solution is constructed for (1.1.4) on the slab in § 4.2 , on the unit disk in § 4.3 and on a general two dimensional geometry in § 4.4. In each case a two term asymptotic expansion is obtained and careful use of logarithmic switchback terms is required. Good agreement with numerical calculations is observed. A interesting observation is that the perturbation −δ∆2u changes the singularity behaviour of the limiting solution from u ∼ −1 + r2/3 in the membrane case (1.1.2) to u ∼ −1 + αr2 log r in the biharmonic case (1.1.4). We conjecture that a necessary condition for a perturbation of (1.1.2) to destroy the infinite fold point structure is that it must alter the singularity of the limiting solution. The analysis of § 4.3.2 deals with the construction of the radially symmetric maximal solution branch for the annulus problem (1.1.6). An added complication in this problem is that the point of maximum deflection, or the concentration point, is not known a priori. This point is determined along with the construction of the singular solution as part of a free boundary problem. 114 4.5. Conclusions Finally in § 4.4, the problem of constructing the maximal solution branch for the pure biharmonic MEMS problem (4.4.1) is considered. Knowledge of a Neumann Green’s function, defined in (4.4.2), is key to the analysis. Under the assupmtion that the singular solution concentrates at some x0 ∈ Ω as u(x0) → −1, the maximal branch is constructed in terms of the Regular Part of the Green’s function (4.4.2) and conditions on the point x0 are established. This Chapter forms the basis of the paper [30] titled Asymptotics of some nonlin- ear eigenvalue problems for a MEMS capacitor: Part II: Fold point Multiple Solutions and Singular Asymptotics under consideration for The European Journal of Applied Mathematics. 115 Chapter 5 Persistence in Patchy Domains In this chapter the indefinite weight eigenvalue problem ∆φ+ λm(x)φ = 0 , x ∈ Ω ; ∂nφ = 0 , x ∈ ∂Ω ; ∫ Ω φ2 dx = 1 (5.0.1) is studied. Equation (5.0.1) has been studied in the context of mathematical ecology by many authors (c.f. [45], [46], [59], [60] ). The function m(x) appearing in (5.0.1) represents the local per capita growth rate of a species evolving in Ω and is tailored to represent the variable quality of the habitat Ω. The principal eigenvalue λ of (5.0.1), corresponding to a positive eigenfunction, is known as the persistence threshold. For a species with density u(x, t) evolving according to the Logistic equation ut = D∆u+ u [m(x)− u] , x ∈ Ω ; ∂nu = 0 , x ∈ ∂Ω ; (5.0.2a) u(x, 0) = u0(x) ≥ 0 , x ∈ Ω, (5.0.2b) it was shown in [59] that if 1/D > λ, then u(x, t) → u∗ as t → ∞ where u∗ is a non- negative equilibrium of (5.0.2). When 1/D < λ, u(x, t) → 0 as t → ∞ indicating that the species has become extinct. Therefore, when λ is as small as possible, the species whose density is represented by u(x, t), persists for the largest range of diffusivities D. An interesting question to ask is, among all m(x) for which ∫ Ωm(x) is fixed, which m(x) generates the smallest positive λ where λ is the principal eigenvalue of (5.0.1)? This question has been addressed by many authors over the last 20 years and while a large quantity of information is now known about the optimal functions m(x), the problem remains largely open. Two key established results are the following. The first of which, due to Senn and Hess [68], states that the principal eigenvalue λ of (5.0.1) is positive if and only if ∫ Ωm < 0 and m(x) is positive on some subset of Ω with positive measure. The second key result of [59] states that, the optimal m(x) must be of bang-bang type, i.e. a piecewise constant. In this chapter, the principal eigenvalue of (5.0.1) is constructed and optimized for 116 Chapter 5. Persistence in Patchy Domains a particular class of patchy m(x) given by mε(x) =   mj/ε 2 , x ∈ Ωεj , j = 1, . . . , n , −mb , x ∈ Ω\ ⋃n j=1Ωεj . (5.0.3) Here Ωεj ≡ {x | |x− xj| ≤ ερj ∩ Ω}, so that the patches Ωεj are the portions of the circular disks of radius ερj that are strictly inside Ω. The constant mj is the local growth rate of the jth patch, with mj > 0 for a favourable habitat and mj < 0 for a non-favourable habitat. The constant mb > 0 is the background bulk decay rate for the unfavourable habitat. In terms of this patch arrangement, the condition ∫ Ωmdx < 0 is asymptotically equivalent for ε→ 0 to ∫ Ω mdx = −mb|Ω|+ π 2 n∑ j=1 αjmjρ 2 j +O(ε2) < 0 . (5.0.4) To enforce the condition that m(x) is positive on some subdomain of Ω with positive measure, at least one mj > 0 for some j. The parameters of the problem must always be chosen so that these conditions hold. Based on this specific but relatively general class of m(x), the eigenvalue problem (5.0.1) is solved with the method of matched asymptotic expansion in the limit ε→ 0. In § 5.1, λ is calculated form(x) corresponding to a single positive patch which is located either completely interior to Ω, or centred on a boundary point of Ω. It is observed that a patch centred on the boundary point of Ω always generates a smaller persistence threshold λ than that of a patch centred on an interior point of Ω. In § 5.2, the persistence threshold is calculated asymptotically for a configuration of n patches corresponding to m(x) given in (5.0.3). The two term asymptotic expansion λε = νµ0 + ν 2µ1(x1, . . . , xn) + · · · ν = −1 log ε (5.0.5) is obtained for the persistence threshold λ. The second order term µ1, which has an explicit dependence on the centres of the patches, is determined in terms the Neumann Green’s function (5.1.19). In § 5.3, the persistence threshold is optimized with respect to patch fragmenta- tion and location. Interestingly, a large amount of information concerning the optimal arrangement of patches is contained in the leading order term of the expansion of λ. In § 5.3.2, several qualitative results are established. It is observed that a favourable patch centred on the boundary of Ω always produces a lower persistence threshold than a patch centred at an interior point of Ω. If the boundary of the domain has corners ( 117 5.1. Determination of The Persistence Threshold for One Patch e.g. squares, triangles ), the corner of smallest angle is the most preferred location for a favourable patch. It is also shown that when one favourable interior patch is split into two favourable interior patches, the persistence threshold λ is always increased. This indicates that fragmentation of resources is generally not advantageous to the species. It is observed that splitting a favourable interior patch into a favourable interior and a favourable boundary patch is only advantageous if the boundary patch is sufficiently strong. Finally in § 5.3.3, the degenerate case where the leading order term in λ is optimized for multiple configurations is considered. In this scenario, optimization of the second order term, µ1 is investigated and some instructive examples are constructed. 5.1 Determination of The Persistence Threshold for One Patch In this section, the method of matched asymptotic expansions is used to derive a two- term asymptotic expansion for the positive principal eigenvalue λ of (1.2.3) for the case of one localized favourable habitat centred at either a point interior to Ω or on ∂Ω. 5.1.1 A Single Interior Patch First, the case where one interior circular patch centred at x0 ∈ Ω, with dist(x0, ∂Ω)≫ O(ε) is calculated. The positive principal eigenvalue λ > 0 and the corresponding eigenfunction φ > 0 of ∆φ+ λmε(x)φ = 0 , x ∈ Ω; ∂nφ = 0 , x ∈ ∂Ω ; ∫ Ω φ2 dx = 1 , (5.1.1a) are calculated asymptotically in the small patch radius limit ε → 0, where the growth rate function mε(x) is defined as mε(x) =   m+/ε 2 , x ∈ Ωε0 , −mb, x ∈ Ω\Ωε0 . (5.1.1b) Here the patch Ωε0 is the circular disk Ωε0 ≡ {x | |x− x0| ≤ ε }. In (5.1.1b), m+ > 0 is the local growth rate of the favourable habitat, while mb > 0 gives the background bulk decay rate for the unfavourable habitat. The condition ∫ Ωmdx < 0 for the existence of a positive principal eigenvalue is 118 5.1. Determination of The Persistence Threshold for One Patch asymptotically equivalent to ∫ Ω mdx = −mb|Ω|+ πm+ +O(ε2) < 0 , (5.1.2) in the limit ε→ 0. It is assumed that the constants mb and m+ are chosen so that this condition holds. The positive principal eigenvalue λ of (5.1.1) is expanded as λ ∼ µ0ν + µ1ν2 + · · · , ν = −1/ log ε , (5.1.3) for some coefficients µ0 and µ1 to be found. In the outer region, defined away from an O(ε) neighbourhood of x0, we expand the corresponding eigenfunction as φ ∼ φ0 + νφ1 + ν2φ2 + · · · . (5.1.4) Upon substituting (5.1.3) and (5.1.4) into (5.1.1), φ0 is observed to be a constant with the normalization condition ∫ Ω φ 2 0 dx = 1 yielding φ0 = |Ω|−1/2, where |Ω| is the area of Ω. In addition, φ1 and φ2 are found to satisfy ∆φ1 = µ0mbφ0 , x ∈ Ω\{x0} ; ∂nφ1 = 0 , x ∈ ∂Ω ;∫ Ω φ1 dx = 0 , (5.1.5a) ∆φ2 = µ1mbφ0 + µ0mbφ1 , x ∈ Ω\{x0} ; ∂nφ2 = 0 , x ∈ ∂Ω ;∫ Ω ( φ21 + 2φ0φ2 ) dx = 0 . (5.1.5b) The matching of φ1 and φ2 to an inner solution defined in an O(ε) neighborhood of the patch at x0, as done below, will yield singularity conditions for φ1 and φ2 as x→ x0. In the inner region near the patch centred at x0 the local variables y and ψ are introduced by y = ε−1(x− x0) , ψ(y) = φ(x0 + εy) . (5.1.6) With these new variables, (5.1.1) becomes ∆ψ =   −λm+ψ , |y| < 1 , O(ε2) , |y| > 1 . (5.1.7) The inner approximation to the eigenfunction is expanded as ψ ∼ ψ0 + νψ1 + ν2ψ2 + · · · , ν = −1/ log ε . (5.1.8) 119 5.1. Determination of The Persistence Threshold for One Patch where to leading order ψ0 is observed to be an unknown constant while ψ1 and ψ2 satisfy ∆ψk =   Fk , |y| ≤ 1 , 0 , |y| ≥ 1 . (5.1.9a) Here Fk for k = 1, 2 is defined by F1 = −µ0m+ψ0 , F2 = −µ0m+ψ1 − µ1m+ψ0 . (5.1.9b) The solution ψ1 to (5.1.9) is calculated to be ψ1 =   A1ρ 2/2 + ψ̄1 , ρ ≤ 1 , A1 log ρ+ A1 2 + ψ̄1 , ρ ≥ 1 , (5.1.10a) where ρ = |y|. Here ψ̄1 is an unknown constant and A1 is given by A1 = F1 2 = −1 2 µ0m+ψ0 . (5.1.10b) Additionally, the solution ψ2 is observed to have far-field behaviour ψ2 ∼ A2 log ρ+O(1) , as ρ→∞ , A2 ≡ ∫ 1 0 F2 ρ dρ . (5.1.11a) The value of A2 is determined by using (5.1.10) and (5.1.9b) for F2 to obtain A2 = −µ0m+ ∫ 1 0 ( A1 ρ2 2 + ψ̄1 ) ρ dρ− 1 2 µ1m+ψ0 = A1 ψ0 ( A1 4 + ψ̄1 + µ1 µ0 ψ0 ) . (5.1.11b) The matching condition is that the near-field behaviour as x → x0 of the outer repre- sentation of the eigenfunction must agree asymptotically with the far-field behaviour of the inner eigenfunction as |y| = ε−1|x− x0| → ∞, so that φ0 + νφ1 + ν 2φ2 + · · · ∼ ψ0 + νψ1 + ν2ψ2 + · · · . (5.1.12) Upon using the far-field behaviour of ψ1 and ψ2, as given in (5.1.10) and (5.1.11) respectively, (5.1.12) becomes φ0 + νφ1 + ν 2φ2 + · · · ∼ ψ0 +A1 + ν ( A1 log |x− x0|+ A1 2 + ψ̄1 +A2 ) + ν2 (A2 log |x− x0|+O(1)) . (5.1.13) 120 5.1. Determination of The Persistence Threshold for One Patch Matching at leading order gives the following condition on the constants φ0 and ψ0 φ0 = ψ0 +A1 . (5.1.14) The O(ν) term in the matching condition (5.1.13), provides the singularity condition φ1 ∼ A1 log |x− x0|+ A1 2 + ψ̄1 +A2 , as x→ x0 , (5.1.15) which together with (5.1.5a) completes the specification of φ1. Note that the singularity behaviour in (5.1.15) specifies both the regular and singular part of a Coulomb singularity. Consequently, this singularity structure provides one constraint relating A1, A2, and ψ̄1. The problem for φ1 can be written in terms of the Dirac distribution as ∆φ1 = µ0mbφ0 + 2πA1δ(x− x0) , x ∈ Ω ; ∂nφ1 = 0 , x ∈ ∂Ω . (5.1.16) An application of the divergence theorem then yields A1 = − 1 2π (µ0mb|Ω|φ0) . (5.1.17) Next, the solution to (5.1.16) is written in terms of the Neumann Green’s function G(x;x0) as φ1 = −2πA1G(x;x0) = µ0mb|Ω|φ0G(x;x0) . (5.1.18) Here G(x;x0) is the unique solution to ∆G = 1 |Ω| − δ(x− x0) , x ∈ Ω ; ∂nG = 0 , x ∈ ∂Ω ; ∫ Ω Gdx = 0 , (5.1.19a) G(x;x0) ∼ − 1 2π log |x− x0|+R(x0;x0) , as x→ x0 , (5.1.19b) where R(x0;x0) is the regular part of G(x;x0) at x = x0. By expanding φ1 in (5.1.18) as x → x0 and equating the non-singular part of the resulting expression with that of (5.1.15), the following condition is obtained − 2πA1R(x0;x0) = A1 2 + ψ̄1 +A2 . (5.1.20) From the O(ν2) terms in matching condition (5.1.13) the singularity behaviour φ2 ∼ A2 log |x − x0| as x → x0 is obtained where φ2 is the solution to (5.1.5b). In terms of 121 5.1. Determination of The Persistence Threshold for One Patch the Dirac mass, this problem for φ2 can be written as ∆φ2 = µ1mbφ0+µ0mbφ1+2πA2δ(x−x0) , x ∈ Ω ; ∂nφ2 = 0 , x ∈ ∂Ω , (5.1.21) with normalization condition ∫ Ω ( φ21 + 2φ0φ2 ) dx = 0. The divergence theorem, to- gether with ∫ Ω φ1 dx = 0, then yields that 2πA2 = −µ1mb|Ω|φ0 . (5.1.22) The leading-order eigenvalue correction µ0 is obtained by combining (5.1.14) and (5.1.17), together with using A1 = −µ0m+ψ0/2 from (5.1.10b) to see that φ0 = πm+ |Ω|mbψ0 , φ0 = ( 1− µ0m+ 2 ) ψ0 . (5.1.23) Now rearranging for µ0, µ0 = 2 m+ [ 1− πm+|Ω|mb ] , ψ0 = |Ω|mb πm+ φ0 , φ0 = |Ω|−1/2 . (5.1.24) Since ∫ Ωmdx < 0, then m+π/(|Ω|mb) < 1 from (5.1.2). Consequently, it follows from (5.1.24) that µ0 > 0. Combining (5.1.17) and (5.1.22) reveals that the ratio A2/A1 has value A2/A1 = µ1/µ0 which in turn allows ψ̄1 and the eigenvalue correction µ1 to be determined from (5.1.20) and (5.1.11b) ψ̄1 = −A1 4 , µ1 = − ( 1 4 + 2πR(x0;x0) ) µ0 . (5.1.25) Finally, a two-term expansion for the eigenfunction in the outer region is obtained from (5.1.4) by using (5.1.18) for φ1. The corresponding two-term inner approximation to the eigenfunction is given by (5.1.8), where ψ1 is given in (5.1.10) with ψ̄1 = −A1/4. In Summary, Principal Result 5.1: In the limit of small patch radius, ε→ 0, the positive principal eigenvalue λ of (5.1.1) has the following two-term asymptotic expansion in terms of the logarithmic gauge function ν = −1/ log ε: λ = µ0ν − µ0ν2 [ 1 4 + 2πR(x0;x0) ] +O(ν3) ; µ0 ≡ 2 m+ [ 1− πm+|Ω|mb ] . (5.1.26a) A two-term asymptotic expansion for the corresponding eigenfunction in the outer region |x− x0| ≫ O(ε) is φ ∼ φ0 (1 + νµ0mb|Ω|G(x;x0)) . (5.1.26b) 122 5.1. Determination of The Persistence Threshold for One Patch Here G(x;x0) is the Neumann Green’s function of (5.1.19) with regular part R(x0;x0). The corresponding inner approximation to the eigenfunction, with y = ε−1(x− x0) and ρ = |y| = O(1), is ψ ∼ mb|Ω| m+π φ0 ( 1− µ0m+ 2 νψ̃1(ρ) ) , (5.1.26c) where φ0 = |Ω|−1/2, and ψ̃1(ρ) is defined by ψ̃1(ρ) ≡   ρ2/2− 1/4 , ρ ≤ 1 , log ρ+ 1/4 , ρ ≥ 1 . (5.1.26d) The eigenvalue problem (5.1.1) is explicitly solvable only for the special case where Ω is the unit disk with a circular patch of radius ε centred at the origin. For this special case, the solution to (5.1.1), which is continuous across the patch boundary r = ε, is φ =   C [ I0 (√ λmbr )− I ′0 (√ λmb ) K ′0 (√ λmb )K0 (√λmbr) ] , ε ≤ r ≤ 1, C [ I0 (√ λmbε )− I ′0 (√ λmb ) K ′0 (√ λmb )K0 (√λmbε) ] J0 (√ λm+r/ε ) J0 (√ λm+ ) , 0 ≤ r ≤ ε. (5.1.27) Here I0(z) and K0(z) are the modified Bessel functions of the first and second kind of order zero. By imposing that φ is smooth across r = ε, and recalling that J ′0(z) = −J1(z), I ′0(z) = I1(z) and K ′0(z) = −K1(z), the following transcendental equation for λ is obtained: ε √ mb m+ J0 (√ λm+ ) J1 (√ λm+ ) = K0 (√ λmbε ) I1 (√ λmb ) +K1 (√ λmb ) I0 (√ λmbε ) K1 (√ λmbε ) I1 (√ λmb )−K1 (√λmb) I1 (√λmbε) . (5.1.28) The first positive root of (5.1.28) is the positive principle eigenvalue of (5.1.1). For ε→ 0, we expand this root as λ = µ0ν + µ1ν 2 + · · · , ν ≡ −1/ log ε . (5.1.29a) By using well-known asymptotic formulae for the Bessel and Modified Bessel functions of small argument, we substitute (5.1.29a) into (5.1.28), and equate coefficients in powers of ν to obtain that µ0 = 2 m+ [ 1− m+ mb ] , µ1 = µ0 2 . (5.1.29b) 123 5.1. Determination of The Persistence Threshold for One Patch 0 0.05 0.1 0.15 0 0.1 0.2 0.3 0.4 0.5 0.6 ε λ* Figure 5.1: Plot of the two-term asymptotic expansion (dashed curve) (5.1.29) for λ versus ε when Ω is the unit disk with a concentric circular patch of radius ε centred at the origin. The solid curve is the eigenvalue λ as obtained from the exact transcendental relation (5.1.28). The parameter values are mb = 2 and m+ = 1. For the special case where Ω is the unit disk containing a circular patch of radius ε, we need only substitute |Ω| = π and R(0; 0) = −3/(8π), obtained from Appendix B of [31], into (5.1.26a). The resulting two-term expansion for λ agrees with (5.1.29). In Fig. 5.1 a very favourable comparison is displayed between the two-term expansion (5.1.29) for λ and the corresponding exact result obtained by finding the first positive root of (5.1.28) numerically. 5.1.2 A Single Boundary Patch Next, the case where the centre x0 of the circular patch is on the boundary of the domain ∂Ω where ∂Ω is piecewise differentiable, but may have corners with nonzero angle. The boundary patch Ωε0 ≡ {x | |x − x0| ≤ ερ0 ∩ Ω} with x0 ∈ ∂Ω is the portion of a circular disk of radius ερ0 that is strictly contained within Ω. In the limit ε → 0, and for x− x0 = O(ε), define πα0 to be the angular fraction of the circular patch that is contained within Ω. More specifically, α0 = 1 whenever x0 is at a smooth point of ∂Ω, and α0 = 1/2 when x0 is at a π/2 corner of ∂Ω. The eigenvalue problem associated with this boundary patch is ∆φ+ λmε(x)φ = 0 , x ∈ Ω; ∂nφ = 0 , x ∈ ∂Ω ; ∫ Ω φ2 dx = 1 , (5.1.30a) 124 5.1. Determination of The Persistence Threshold for One Patch where mε(x) is defined as mε(x) =   m+/ε 2 , x ∈ Ωε0 , −mb , x ∈ Ω\Ωε0 . (5.1.30b) The condition ∫ Ωmdx < 0 is asymptotically equivalent when ε→ 0 to∫ Ω mdx = −mb|Ω|+ α0π 2 ( m+ρ 2 0 ) +O(ε2) < 0 . (5.1.31) The parameters of the problem are assumed to be chosen such that this condition on∫ Ωmdx holds. Since the asymptotic calculation of λ for a boundary patch is similar to that for the interior patch case, we mainly highlight the new features that are required in the analysis. First, λ is expanded in terms of ν = −1/ log ε as in (5.1.3). In the outer region, defined for |x − x0| ≫ O(ε), the expansion of the outer solution, as in (5.1.4), is used to obtain that φ0 is a constant, and that φ1 and φ2 satisfy (5.1.5a) and (5.1.5b) in Ω, respectively, with ∂nφk = 0 for x ∈ ∂Ω\{x0} for k = 1, 2. Since the expansion of the inner solution is again in powers of ν = −1/ log ε as in (5.1.8), the effect of the curvature of the domain boundary near x = x0 can be neglected to any power of ν, provided that this curvature is finite. Consequently, when x0 is at a smooth point of ∂Ω, the inner region near x = x0 can be approximated by the tangent line to ∂Ω through x = x0. Alternatively, when x0 is at corner point of ∂Ω, the inner region is the angular wedge of angle πα0 bounded by the intersection of the one-sided tangent lines to ∂Ω at x = x0. The local variable y = ε −1(x − x0) is used so that the inner region is the angular wedge β0 < arg y ≤ α0π + β0 for some β0. The favourable habitat is the circular patch |y| ≤ ρ0 that lies within this wedge. Since the no-flux boundary conditions ∂nψ = 0 holds on the two sides of the wedge, a local radially symmetric inner solution is established within the angular wedge. Therefore, in the inner region, the solution is expanded as in (5.1.8) to obtain that ψ0 is a constant, and that ψk for k = 1, 2 satisfies ∆ψk =   Fk , |y| ≤ ρ0 , β0 ≤ arg y ≤ πα0 + β0 , 0 , |y| ≥ ρ0 , β0 ≤ arg y ≤ πα0 + β0 . (5.1.32) 125 5.1. Determination of The Persistence Threshold for One Patch Here Fk for k = 1, 2 are defined in (5.1.9b). The solution for ψ1, with ρ = |y|, is ψ1 =   A1 ( ρ2 2ρ20 ) + ψ̄1 , 0 ≤ ρ ≤ ρ0 , β0 ≤ arg y ≤ πα0 + β0 , A1 log ( ρ ρ0 ) + A1 2 + ψ̄1 , ρ ≥ ρ0 , β0 ≤ arg y ≤ πα0 + β0 , (5.1.33) where ψ̄1 is an unknown constant and A1 = F1ρ20/2. For ψ2, we obtain that ψ2 ∼ A2 log ρ as ρ → ∞. The calculation of A2 proceeds exactly as in (5.1.11b) to obtain that A1 = −µ0 2 m+ρ 2 0ψ0 , A2 = A1 ψ0 ( A1 4 + ψ̄1 + µ1 µ0 ψ0 ) . (5.1.34) The matching condition between the outer solution as x→ x0 and the inner solution for |y| = ε−1|x− x0| → ∞ is given by (5.1.12). Upon using (5.1.33) for ψ1 when ρ≫ 1, together with ψ2 ∼ A2 log ρ for ρ≫ 1, the condition φ0 + νφ1 + ν 2φ2 + · · · ∼ ψ0 +A1 + ν ( A1 log |x− x0| −A1 log ρ0 + A1 2 + ψ̄1 +A2 ) + ν2 (A2 log |x− x0|+O(1)) . (5.1.35) is established. The leading order matching condition from (5.1.35) is that φ0 = ψ0 +A1 . (5.1.36) Appending the singularity condition established from the O(ν) terms of (5.1.35) to problem (5.1.5a), gives the following complete specification of φ1 ∆φ1 = µ0mbφ0 , x ∈ Ω ; ∂nφ1 = 0 , x ∈ ∂Ω\{x0} ; ∫ Ω φ1 dx = 0 , (5.1.37a) φ1 ∼ A1 log |x− x0| −A1 log ρ0 + A1 2 + ψ̄1 +A2 , as x→ x0 . (5.1.37b) The singularity condition from the O(ν2) terms in (5.1.35) are appended to the problem for φ2 (5.1.5b) to give ∆φ2 = µ1mbφ0 + µ0mbφ1 , x ∈ Ω ; ∂nφ2 = 0 , x ∈ ∂Ω\{x0} ; (5.1.38a)∫ Ω ( φ21 + 2φ0φ2 ) dx = 0; φ2 ∼ A2 log |x− x0|+O(1) , as x→ x0 . (5.1.38b) The divergence theorem is now applied to (5.1.37) over Ω\Ωσ, where Ωσ is a wedge of angle πα0 and small radius σ ≪ 1 centred at x0 ∈ ∂Ω. Imposing the singularity 126 5.1. Determination of The Persistence Threshold for One Patch condition (5.1.37b) on |x− x0| = σ and taking the limit σ → 0, the relationship µ0mb|Ω|φ0 = −α0πA1 (5.1.39) is established. In a similar way, applying the divergence theorem to (5.1.38), noting that ∫ Ω φ1 dx = 0, gives the following equality involving A2 µ1mb|Ω|φ0 = −α0πA2 . (5.1.40) Combining (5.1.39) and (5.1.40) gives that A2/A1 = µ1/µ0, which yields ψ̄1 = −A1/4 from the equation for A2 in (5.1.34). Then, by combining (5.1.36), (5.1.34) for A1, and (5.1.39), the leading order correction terms ψ0 = 2mb|Ω| α0πm+ρ20 φ0 , µ0 = 2 m+ρ20 [ 1− α0πm+ρ 2 0 2mb|Ω| ] (5.1.41) are established. Since ∫ Ωmdx < 0 from (5.1.31), it follows that µ0 > 0 in (5.1.41). Equation (5.1.37) admits the compact representation φ1 = −α0πA1Gs(x;x0) (5.1.42) in terms of the surface Neumann Green’s function Gs(x;x0), defined as the unique solution of ∆Gs = 1 |Ω| , x ∈ Ω ; ∂nGs = 0 , x ∈ ∂Ω\{x0} ; ∫ Ω Gs dx = 0 , (5.1.43a) Gs(x;x0) ∼ − 1 α0π log |x− x0|+Rs(x0;x0) , as x→ x0 ∈ ∂Ω . (5.1.43b) Here |Ω| is the area of Ω, and Rs(x0;x0) is the regular part of the surface Neumann Green’s function at x = x0. By expanding φ1 as x → x0 using (5.1.43b), the log |x − x0| term matches auto- matically whereas matching the nonsingular part of φ1 as x → x0 gives the condition (5.1.37b) to obtain − α0πA1Rs(x0;x0) = −A1 log ρ0 + A1 2 + ψ̄1 +A2 . (5.1.44) Using ψ̄1 = −A1/4 and A2/A1 = µ1/µ0 in (5.1.44), and solving for µ1 gives µ1 = µ0 [ log ρ0 − 1 4 − α0πRs(x0;x0) ] . (5.1.45) 127 5.2. The Persistence Threshold for Multiple Patches In summary: Principal Result 5.2: In the limit of small boundary patch radius, ε → 0, a two- term asymptotic expansion for the positive principal eigenvalue λ of (5.1.30) in terms of ν = −1/ log ε is λ = µ0ν − µ0ν2 [ 1 4 + α0πRs(x0;x0)− log ρ0 ] +O(ν3) ; µ0 ≡ 2 m+ρ20 [ 1− α0πm+ρ 2 0 2|Ω|mb ] . (5.1.46a) A two-term asymptotic expansion for the corresponding eigenfunction in the outer region |x− x0| ≫ O(ε) is φ ∼ φ0 (1 + νµ0mb|Ω|Gs(x;x0)) . (5.1.46b) Here Gs(x;x0) is the surface Neumann Green’s function of (5.1.43) with regular part Rs(x0;x0). The implication of Principal Results 5.1 and 5.2 for the determination of the persis- tence threshold is discussed in § 5.3.1. 5.2 The Persistence Threshold for Multiple Patches In this section the analysis of § 5.1 is generalized to treat the case of an arbitrary but fixed number n of circular patches, each of which is centred either inside Ω or on ∂Ω. To this end, the positive principal eigenvalue of ∆φ+ λmε(x)φ = 0 , x ∈ Ω; ∂nφ = 0 , x ∈ ∂Ω ; ∫ Ω φ2 dx = 1 , (5.2.1a) is calculated asymptotically where the growth rate function mε(x) is defined by mε(x) =   mj/ε 2 , x ∈ Ωεj , j = 1, . . . , n , −mb , x ∈ Ω\ ⋃n j=1Ωεj . (5.2.1b) Here Ωεj ≡ {x | |x− xj | ≤ ερj ∩ Ω}, so that the patches Ωεj are the portions of the circular disks of radius ερj that are strictly inside Ω. The constant mj is the local growth rate of the jth patch, with mj > 0 for a favourable habitat and mj < 0 for a non-favourable habitat. The constant mb > 0 is the background bulk decay rate for the unfavourable habitat. In terms of this patch arrangement, the condition ∫ Ωmdx < 0 is 128 5.2. The Persistence Threshold for Multiple Patches asymptotically equivalent for ε→ 0 to ∫ Ω mdx = −mb|Ω|+ π 2 n∑ j=1 αjmjρ 2 j +O(ε2) < 0 . (5.2.2) The parameters of the problem must always be chosen so that this condition holds. The patches are assumed to be well-separated in the sense mentioned in § 1.2. The parameters in the growth rate are the centres x1, . . . , xn of the circular patches, their radii ερ1, . . . , ερn, the local growth rates m1, . . . ,mn, the angular fractions πα1, . . . , παn of the circular patches that are contained in Ω, and the constant bulk growth rate mb. Recall that αj = 2 whenever xj ∈ Ω, αj = 1 when xj ∈ ∂Ω and xj is a point where ∂Ω is smooth, and αj = 1/2 when xj ∈ ∂Ω is at a π/2 corner of ∂Ω, etc. To asymptotically analyze (5.2.1), both the Neumann Green’s function and the surface Neumann Green’s function are used. As such, the following definition for a generalized modified Green’s function Gm(x;xj) is useful, Gm(x;xj) ≡   G(x;xj) , xj ∈ Ω , Gs(x;xj) , xj ∈ ∂Ω . (5.2.3a) Here G(x;xj) is the Neumann Green’s function of (5.1.19), and Gs(x;xj) is the surface Neumann Green’s function of (5.1.43). Therefore, the local behaviour of Gm(x;xj) is Gm(x;xj) ∼ − 1 αjπ log |x− xj|+Rm(xj ;xj) , as x→ xj ; Rm(xj ;xj) ≡   R(xj;xj) , xj ∈ Ω , Rs(xj;xj) , xj ∈ ∂Ω . (5.2.3b) Here R(xj ;xj) and Rs(xj ;xj) are the regular part of the Neumann Green’s function (5.1.19) and the surface Neumann Green’s function (5.1.43), respectively. To derive a two-term expansion for the positive principal eigenvalue of (5.2.1), λ is expanded as in (5.1.3), and φ as in (5.1.4). Upon substituting (5.1.3) and (5.1.4) into (5.2.1), φ0 = |Ω|−1/2 is observed to be a constant, and that φ1 and φ2 satisfy ∆φ1 = µ0mbφ0 , x ∈ Ω\ΩI ; ∂nφ1 = 0 , x ∈ ∂Ω\ΩB ;∫ Ω φ1 dx = 0 , (5.2.4a) ∆φ2 = µ1mbφ0 + µ0mbφ1 , x ∈ Ω\ΩI ; ∂nφ2 = 0 , x ∈ ∂Ω\ΩB ;∫ Ω ( φ21 + 2φ0φ2 ) dx = 0 . (5.2.4b) 129 5.2. The Persistence Threshold for Multiple Patches In the inner region, near the jth patch, local variables y = ε−1(x − xj) and ψ(y) = φ(xj + εy) are introduced. In this region, ψ is expanded for y = O(1) with ψ ∼ ψ0j + νψ1j + ν2ψ2j + · · · , (5.2.5) where ψ0j is a constant to be determined. For an interior patch with xj ∈ ΩI , ψkj for k = 1, 2 satisfy ∆ψkj =   Fkj , |y| ≤ ρj , 0 , |y| ≥ ρj , (5.2.6) where F1j = −µ0mjψ0j and F2j = −µ0mjψ1j − µ1mjψ0j . The solution for ψ1j , with ρ = |y|, is ψ1j =   A1j ( ρ2 2ρ2j ) + ψ̄1j , 0 ≤ ρ ≤ ρj , A1j log ( ρ ρj ) + A1j 2 + ψ̄1j , ρ ≥ ρj , (5.2.7) where ψ̄1j is an unknown constant. In addition, ψ2j ∼ A2j log ρ as ρ → ∞. The divergence theorem is used to calculate A1j and A2j from (5.2.6) to obtain A1j = −µ0 2 mjρ 2 jψ0j , A2j = A1j ψ0j ( A1j 4 + ψ̄1j + µ1 µ0 ψ0j ) . (5.2.8) For a boundary patch, for which xj ∈ ΩB, then (5.2.6) holds in the wedge βj < arg(y) < βj + παj , for some βj and 0 < αj < 2. For this boundary case, the constants A1j and A2j are also given by (5.2.8). The matching condition between the outer solution as x→ xj and the inner solution as |y| = ε−1|x− xj | → ∞ is φ0 + νφ1 + ν 2φ2 + · · · ∼ ψ0j +A1j + ν ( A1j log |x− xj | −A1j log ρj + A1j 2 + ψ̄1j +A2j ) + ν2 (A2j log |x− xj |+O(1)) . (5.2.9) The leading-order matching condition from (5.2.9) yields φ0 = ψ0j +A1j , j = 1, . . . , n . (5.2.10) From the O(ν) terms in (5.2.9), the following singularity behaviour is obtained for φ1 130 5.2. The Persistence Threshold for Multiple Patches as x→ xj φ1 ∼ A1j log |x− xj | −A1j log ρj + A1j 2 + ψ̄1j +A2j , as x→ xj . (5.2.11) In addition, from the O(ν2) terms in (5.2.9), we conclude that φ2 ∼ A2j log |x− xj |+O(1) , as x→ xj . (5.2.12) Next, by using the divergence theorem on the solution φ1 to (5.2.4a) with singular behaviour (5.2.11), the following equality is established µ0mb|Ω|φ0 = −π n∑ j=1 αjA1j . (5.2.13) Similarly, the divergence theorem applied to (5.2.4b) with singular behaviour (5.2.12), and noting ∫ Ω φ1 dx = 0, yields µ1mb|Ω|φ0 = −π n∑ j=1 αjA2j . (5.2.14) Combining (5.2.10) and (5.2.8) allows the values of A1j and ψ0j to be isolated as ψ0j = 2φ0 2−mjρ2jµ0 , A1j = − mjρ 2 jµ0φ0 2−mjρ2jµ0 , j = 1, . . . , n . (5.2.15) From (5.2.13), together with (5.2.15) for A1j , the leading-order eigenvalue correction µ0 is observed to satisfy the algebriac equation mb|Ω| π = n∑ j=1 αjmjρ 2 j 2−mjρ2jµ0 . (5.2.16) The properties of this equation are studied below in § 5.3 and will be shown to provide a large amount of information on the optimal configuration of m(x). The value of the next correction, µ1 is now determined in terms of the Neumann Green’s Function G(x;x0) and its regular part R(x;x0), defined in (5.1.19). First, the solution φ1 to (5.2.4a), with singular behaviour (5.2.11), is written in terms of the modified Green’s function Gm(x;xj) of (5.2.3) as φ1 = −π n∑ i=1 αiA1iGm(x;xi) . (5.2.17) 131 5.2. The Persistence Threshold for Multiple Patches Then, by expanding φ1 as x → xj and by using (5.2.3b) for the local behaviour of Gm(x;xj), the condition φ1 ∼ A1j log |x− xj | − παjA1jRmjj +Bj , x→ xj ; Bj ≡ −π n∑ i=1 i6=j αiA1iGmji , (5.2.18) is obtained where Gmji ≡ Gm(xj ;xi). The requirement that the nonsingular terms in (5.2.11) and (5.2.18) agree yields the constraints − παjA1jRmjj +Bj = −A1j log ρj + A1j 2 + ψ̄1j +A2j , j = 1, . . . , n , (5.2.19) whereRmjj ≡ Rm(xj;xj) is the regular part of the generalized modified Green’s function as defined in (5.2.3b). Expressions (5.2.8), (5.2.15), and (5.2.19) can be combined to isolate A2j , then µ1 is determined from (5.2.14) after first solving (5.2.19) for ψ̄1j . Upon substituting the resulting expression for ψ̄1j , together with A1j/ψ0j = −mjρ2jµ0/2 from (5.2.8), into (5.2.8) for A2j , it follows that A2j = − mjρ 2 jµ0 2 ( −A1j 4 − παjA1jRmjj +Bj +A1j log ρj −A2j ) + µ1 µ0 A1j (5.2.20) for each j = 1, . . . , n. Upon solving this equation for A2j , and using (5.2.18) for Bj, the product αjA2j is found to be αjA2j = − mjρ 2 jµ0( 2−mjρ2jµ0 )[− πα2jA1jRmjj + αjA1j log ρj − αjA1j 4 − π n∑ i=1 i6=j αiαjA1iGmji ] + µ1 µ0 ( 2A1jαj 2−mjρ2jµ0 ) . (5.2.21) Next, it is convenient to introduce a new variable κj and to rewrite A1j of (5.2.15) in terms of this variable as κj ≡ √ αjmjρ 2 j 2−mjρ2jµ0 , A1j = −µ0κj√ αj φ0 , j = 1, . . . , n . (5.2.22) It is also convenient to introduce the symmetric n × n Green’s matrix Gm, and the 132 5.2. The Persistence Threshold for Multiple Patches diagonal matrix P, with matrix entries Gmij and Pij defined by Gmij = √αiαjGmij , i 6= j ; Gmjj = αjRmjj ; Pij = 0 , i 6= j ; Pjj = log ρj . (5.2.23) In terms of κj , Gm, P, and the vector κ = (κ1, . . . , κn)t, (5.2.21) readily reduces to αjA2j = −µ20φ0κj [ π (Gmκ)j − (Pκ)j + κj 4 ] − 2µ1κ 2 j mjρ2j φ0 , (5.2.24) where (Pκ)j and (Gmκ)j denote the jth component of the vectors Pκ and Gmκ, respec- tively. Finally, by substituting (5.2.24) into (5.2.14), and solving the expression for µ1 is µ1  mb|Ω| π − n∑ j=1 2κ2j mjρ2j   = µ20 [ κt (πGm − P)κ+ 1 4 κtκ ] . (5.2.25) The left-hand side of (5.2.25) is simplified by using the equation (5.2.16) for µ0 to obtain mb|Ω| π − n∑ j=1 2κ2j mjρ 2 j = n∑ j=1 αj( 2−mjρ2jµ0 )  mjρ2j − 2mjρ2j( 2−mjρ2jµ0 )   , = − n∑ j=1 αj( 2−mjρ2jµ0 )   µ0m2jρ4j( 2−mjρ2jµ0 )   = −µ0κtκ . (5.2.26) This determines µ1 from (5.2.25) in terms of a Rayleigh-type quotient. In summary: Principal Result 5.3: In the limit of small patch radius, ε→ 0, the positive principal eigenvalue λ of (5.2.1) has the following two-term asymptotic expansion in terms of the logarithmic gauge function ν = −1/ log ε: λ = µ0ν − µ0ν2 ( κt (πGm − P)κ κtκ + 1 4 ) +O(ν3) . (5.2.27) Here µ0 > 0 is the first positive root of B(µ0) = 0, where B(µ) is defined by B(µ) ≡ −mb|Ω|+ π n∑ j=1 αjmjρ 2 j 2−mjρ2jµ . (5.2.28) In (5.2.27), κ = (κ1, . . . , κn) t, where κj is defined in (5.2.22), while Gm and P are the n × n matrices as defined in (5.2.23). In addition, a two-term expansion for the outer 133 5.2. The Persistence Threshold for Multiple Patches solution is given by φ ∼ φ0  1 + νπµ0 n∑ j=1 √ αjκjGm(x;xj)   . (5.2.29) Next, equation (5.2.28) is shown to admit a positive root µ0 > 0. Since ∫ Ωmdx < 0 from (5.2.2), it follows that B(0) < 0 from (5.2.28). In addition, B(µ) → +∞ as µ→ 2/(mJρ2J) from below, where mJρ2J is defined by mJρ 2 J = max mj>0 {mjρ2j | j = 1, . . . , n } . (5.2.30) There must be at least one j for whichmj > 0, so that (5.2.30) is attained at some j = J . Moreover, (5.2.28) readily yields that B′(µ) > 0 on 0 < µ < 2/(mJρ2J). Therefore, there exists a unique root µ = µ0 on 0 < µ < 2/(mJρ 2 J) satisfying B(µ0) = 0. The corresponding leading-order eigenfunction in the inner region, ψ0j , satisfies ψ0j > 0 from (5.2.15). Therefore, µ0 is the leading-order term in the asymptotic expansion of the positive principal eigenvalue of (5.2.1). Although the required root to (5.2.28) must in general be computed numerically, there are two special cases where it can be found analytically. In the symmetric case where mj = mc and ρj = ρc for j = 1, . . . , n, then, the root of (5.2.28) is simply µ0 = 2 mcρ2c [ 1− παsmcρ 2 c 2mb|Ω| ] , αs ≡ n∑ j=1 αj . (5.2.31) In addition, if there are only two types of patches, such as mjρ 2 j = mcρ 2 c for j = 1, . . . , n− 1 and mnρ2n, then (5.2.28) reduces to a quadratic equation for µ0, which can be solved explicitly. Note that the asymptotic analysis leading to Principal Result 5.3 has two limitations. First, it is valid only when all interior or boundary patches are well-separated in the sense that |xi − xj| ≫ O(ε) for i 6= j. Second, all interior patches must not be too close to the boundary, in the sense that |x − xj | ≫ O(ε) for xj ∈ ΩI and x ∈ ∂Ω. The easement of either of these two restrictions leads to more complicated inner patch problems that do not appear to be tractable analytically. 134 5.3. Effect of Habitat Fragmentation and Location on Species Persistence 5.3 Effect of Habitat Fragmentation and Location on Species Persistence In this section, the formulae derived in § 5.1 and § 5.2 for the persistence threshold, λ(ε), are used to determine the optimal strategy for distributing a fixed quantity of resources in some domain where favourable and unfavourable patches may already be present. The constraint that the resources being distributed are fixed is expressed mathematically by −mb|Ω|+ π 2 n∑ j=1 αjmjρ 2 j +O(ε2) = ∫ Ω mdx = −K , (5.3.1) where K > 0 is kept constant as mb, or αj , mj , and ρj , for j = 1, . . . , n are varied. 5.3.1 The Persistence Threshold for One Patch Consider first the case of one favourable habitat. For an interior patch of area πε2, recall that λ is given in (5.1.26a) of Principal Result 5.1. For a boundary patch of the same area, we must set πα0ε 2ρ20/2 = πε 2 in (5.1.46a) of Principal Result 5.2. Thus, ρ0 = √ 2/α0, so that (5.1.46a) becomes λ = µ0ν − µ0ν2 [ 1 4 + α0πRs(x0;x0)− 1 2 log ( 2 α0 )] +O(ν3) ; µ0 ≡ α0 m+ [ 1− πm+|Ω|mb ] . (5.3.2) By comparing the leading-order O(ν) terms in (5.3.2) and (5.1.26a), and noting that α0 < 2 for a boundary patch, the following result is established: Principal Result 5.4: For a favourable habitat of area πε2, the positive principal eigenvalue λ is always smaller for a boundary patch than for an interior patch. For a domain boundary with corners, λ is minimized when the boundary patch is centred at the corner with the smallest corner angle πα0. For a domain with smooth boundary, for which α0 = 1 for any x0 ∈ ∂Ω, then λ in (5.3.2) is minimized when the centre x0 of the boundary patch is located at the global maximum of the regular part Rs(x0;x0) of the surface Neumann Green’s function of (5.1.43) on ∂Ω. Principal Result 5.4 shows that for a square, the best choice for the favourable habitat is to concentrate resources near one of the four corners of the square. However, for a domain Ω with a smooth boundary ∂Ω, it is not clear whether the maximization of Rs(x0;x0), as required to minimize λ, has an obvious geometrical interpretation. When Ω is a smooth perturbation of the unit disk, the question of whether the global maximum 135 5.3. Effect of Habitat Fragmentation and Location on Species Persistence of Rs(x0;x0) must necessarily coincide with the global maximum of the curvature of the boundary is addressed. The following result from [62] allows the determination of the critical points of Rs(x0;x0) for domains that are smooth perturbations of the unit disk: Principal Result 5.5: [From [62]]: Let Ω be a smooth perturbation of the unit disk with boundary given in terms of polar coordinates by r = r(θ) = 1 + δσ(θ) , σ(θ) = ∞∑ n=1 (an cos(nθ) + bn sin(nθ)) , δ ≪ 1 . (5.3.3) Let x0 = x0(θ0) = (r0 cos θ0, r0 sin θ0) be a point on the boundary where r0 = 1+ δσ(θ0). For x ∈ ∂Ω we define ρ(θ) ≡ Rs(x;x0) and ρ(θ0) ≡ Rs(x0;x0) , (5.3.4) where Rs(x;x0) is the regular part of the Green’s function defined by Rs(x;x0) = Gs(x;x0) + 1 π log |x− x0| , x ∈ Ω . (5.3.5) Then, for δ ≪ 1, ρ′(θ0) satisfies ρ′(θ0) = δ π ∞∑ n=1 ( n2 + n− 2) (bn cosnθ0 − an sinnθ0) +O(δ2) . (5.3.6) The proof of this result was given in [62]. The following two examples use Principal Result 5.5 to investigate the relationship between the curvature of the boundary and the critical points of the regular part of the surface Green’s function Rs(x;x0). For the first example, take the domain boundary to be r = 1 + δ sin(2θ), so that ρ′(θ0) = 4δπ−1 cos(2θ0) from (5.3.6). For δ ≪ 1, the curvature of the domain boundary is calculated to be κ(θ) = r2 + 2r2θ − rrθθ( r2 + r2θ )3/2 ∼ 1− δ (σ + σθθ) +O(δ2) . (5.3.7) so that for the case r = 1 + δ sin(2θ), ρ(θ) = 2δ π sin(2θ) + C , κ(θ) = 1 + 3δ sin(2θ) , (5.3.8) where C is some constant. For this example the global maxima of ρ and κ over 0 ≤ θ < 2π do coincide, and are attained at θ = π/4 and θ = 5π/4. The following examples uses Principal Result 5.5 to establish the following result: 136 5.3. Effect of Habitat Fragmentation and Location on Species Persistence Principal Result 5.6: The global maximum of Rs(x0, x0) for x0 ∈ ∂Ω does not nec- essarily coincide with the global maximum of the curvature κ(θ) of the boundary of a smooth perturbation of the unit disk. Consequently, for ε→ 0, the persistence threshold λ(ε) from Principal Result 5.2 is not necessarily minimized when the centre of the cir- cular patch is located at the global maximum of the curvature of the smooth boundary ∂Ω. To prove this result, a counterexample is constructed. Take a2 = 1, b3 = b, with an = 0 for n 6= 2 and bn = 0 for n 6= 3 in (5.3.3), so that σ(θ) = cos(2θ) + b sin(3θ) . (5.3.9) For δ ≪ 1, the curvature κ of ∂Ω is calculated from (5.3.7) to be κ = 1 + δ [3 cos(2θ) + 8b sin(3θ)] , κ′(θ) = −6δ [sin(2θ)− 4b cos(3θ)] , κ′′(θ) = −12δ [cos(2θ) + 6b sin(3θ)] . (5.3.10) From (5.3.6) ρ′(θ) and its derivative are calculated to be ρ′(θ) = −4δ π [ sin(2θ)− 5b 2 cos(3θ) ] , ρ′′(θ) = −8δ π [ cos(2θ) + 15b 4 sin(3θ) ] . Therefore, in terms of an unknown constant C, we obtain that ρ(θ) = δ π [ 2 cos(2θ) + 10b 3 sin(3θ) ] + C . (5.3.11) We observe that θ = π/2 and θ = 3π/2 are the only two critical points shared by κ and ρ. The nature of these local extrema depend on the values of κ′′ (π/2) = 12δ(1 + 6b) , ρ′′ (π/2) = 8δ π ( 1 + 15b 4 ) , κ′′ (3π/2) = 12δ(1 − 6b) , ρ′′(3π/2) = 8δ π ( 1− 15b 4 ) . Therefore, when b is chosen to satisfy −4/15 < b < −1/6, then κ has a local maximum while ρ has a local minimum at θ = π/2. Similarly, for this range of b, κ has a local minimum while ρ has a local minimum at θ = 3π/2. Since the only critical points shared by κ and ρ are local minima of ρ, it is concluded that the absolute maximum value of ρ occurs at a point where κ′(θ) 6= 0. Therefore, in general, the point(s) where the absolute maximum value of ρ is attained do not coincide precisely with the maximum curvature of the boundary of the domain. Fig. 5.2(a) 137 5.3. Effect of Habitat Fragmentation and Location on Species Persistence shows a plot of the domain boundary when δ = 0.1 and b = −1/5 while Fig. 5.2(b) plots ρ(θ) − C and κ(θ) − 1 from (5.3.11) and (5.3.10), respectively, for δ = 0.1 and b = −1/5 and demonstrates that the global maxima of ρ and κ − 1 occur at different, but nearby, locations. In §3.3 of [62] a boundary element method (BEM) was formulated and implemented to numerically compute the regular part of the surface Neumann Green’s function for an arbitrary bounded two-dimensional domain with smooth boundary. In Fig. 5.3 a very favourable comparison between full numerical results for ρ(θ) and the perturbation formula (5.3.11) is demonstrated for δ = 0.1 and b = −1/5. The constant C in (5.3.11) was fitted to the full numerical results at θ = 0. This figure provides a numerical validation of the perturbation result (5.3.11). −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x y (a) perturbation of the unit disk 0 1 2 3 4 5 6 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 θ (b) κ(θ)− 1 and ρ(θ) versus θ Figure 5.2: Left figure: plot of the unit disk (dashed curve) and the perturbed unit disk (solid curve) with boundary r = 1 + δ (cos(2θ) + b sin(3θ)), where δ = 0.1 and b = −1/5. Right figure: plot of the curvature perturbation κ(θ)−1 (solid curve) and the regular part ρ(θ) of the surface Neumann Green’s function defined in (5.3.11) (dashed curve) with C = 0. The absolute maximum of κ− 1 and ρ are observed to occur at distinct, but nearby points, as indicated in the figure. It is conjectured that the relationship between the maximum of the boundary cur- vature and the location of the favourable habit that yields the minimum value of λ for a fixed ∫ Ωmdx < 0 is qualitatively similar to that for steady-state bubble-type transition- layer solutions for the Cahn-Hilliard model studied in [50]. In this latter context, it was shown from variational considerations in [50] that the minimal-energy bubble solution attaches orthogonally to the domain boundary at two points, with the global maximum 138 5.3. Effect of Habitat Fragmentation and Location on Species Persistence 0 0.5 1 1.5 2 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3  θ/pi   ρ(θ  )  a nd  κ( θ)− 1  BEM Solution with 128 elements    BEM ρ(θ)  Perturbation ρ(θ)  Perturbation κ(θ) Figure 5.3: Plot of ρ(θ) ≡ R(x0(θ), x0(θ)) computed by the BEM method (heavy solid curve) and the perturbation formula (5.3.11) (dashed curve) versus θ/π for a domain with boundary r = 1 + δ [cos(2θ) + b sin(3θ)] with δ = 0.1 and b = −1/5. The curvature perturbation κ(θ)− 1 is given by the dotted curve. Plot reproduced from [72]. of the boundary curvature located somewhere between these two points. The transition layer associated with this bubble solution is the arc of a circle connecting these two attached boundary points. Similarly, for our boundary patch problem, we expect that for ε small but fixed, the maximum boundary curvature is located somewhere along the curved boundary segment that connects the points where the circular patch intersects the boundary, but is not necessarily at the midpoint of this segment. 5.3.2 Multiple Patches and The Effect Of Fragmentation In this section the effect of both the location and the fragmentation of resources on the leading-order term, µ0, in the asymptotic expansion of λ is considered under the assumption of a fixed value for the constraint in (5.3.1). The analysis below leads to three specific qualitative results. The following simple lemma is central to the derivation of these results: Lemma 5.7: Consider two smooth functions Cold(ζ) and Cnew(ζ) defined on 0 ≤ ζ < µoldm and 0 ≤ ζ < µnewm , respectively, with Cold(0) = Cnew(0) < 0, and Cold(ζ)→ +∞ as ζ → µoldm from below, and Cnew(ζ)→ +∞ as ζ → µnewm from below. Suppose further that there exist unique roots ζ = µold0 and ζ = µ new 0 to Cold(ζ) = 0 and Cnew(ζ) = 0 on the intervals 0 < ζ < µoldm and 0 < ζ < µ new m , respectively. Then, • Case I: If µnewm ≤ µoldm and Cnew(ζ) > Cold(ζ) on 0 < ζ < µnewm , then 139 5.3. Effect of Habitat Fragmentation and Location on Species Persistence µnew0 < µ old 0 . • Case II: If µnewm ≥ µoldm and Cnew(ζ) < Cold(ζ) on 0 < ζ < µoldm , then µnew0 > µ old 0 . The proof of this lemma is a routine exercise in calculus and is omitted. This simple lemma is used to obtain three main qualitative results. First, suppose that the centre of the jth patch of radius ερj with associated angle παj is moved to an unoccupied location, with the new patch having radius ερk and associated angle παk. To satisfy the constraint (5.3.1), the equality αjmjρ 2 j = αkmkρ 2 k must hold. The change in B(ζ), with B(ζ) as defined in (5.2.28), induced by this action is Bnew(ζ)− Bold(ζ) = παkmkρ 2 k 2− ζmkρ2k − παjmjρ 2 j 2− ζmjρ2j = π ( αj αk ) m2jρ 4 jζ (2− ζmjρ2j )(2− ζmkρ2k) (αj − αk) . (5.3.12) Recall from § 5.2 that Bold(ζ) = 0 has a positive root ζ = µold0 on 0 < ζ < µoldm ≡ 2/(mJρ 2 J), where mJρ 2 J was defined in (5.2.30). Assume that αj > αk, e.g. that the centre of an interior patch, for which αj = 2, is moved to a smooth point on the domain boundary, for which αk = 1. First, suppose that the patches are favourable so that mj > 0 and mk > 0. When αj > αk, it follows from the constraint αjmjρ 2 j = αkmkρ 2 k that mkρ 2 k > mjρ 2 j , and so the first vertical asymptote for Bnew(ζ) cannot be larger than that of Bold(ζ). Consequently, we define mKρ 2 K ≡ max{mJρ2J ,mkρ2k}, and from § 5.2 conclude that there is a unique root ζ = µnew0 to Bnew(ζ) = 0 on 0 < ζ < µnewm ≡ 2/(mKρ2K). Since µnewm ≤ µoldm , and (5.3.12) shows that Bnew(ζ) > Bold(ζ) for 0 < ζ < µnewm , then Case I of Lemma 5.7 proves that µnew0 < µ old 0 . Alternatively, for the situation where habitats are unfavourable, so that mj < 0 and mk < 0, then the first vertical asymptotes of Bold(ζ) and Bnew(ζ) must be the same, since these asymptotes are defined only in terms of the favourable patches. For this case, (5.3.12) again shows that Bnew(ζ) > Bold(ζ) for 0 < ζ < 2/(mJρ2J). Case I of Lemma 5.7 then establishes that µnew0 < µ old 0 . Therefore, it is concluded that moving the centre of an interior patch to a point on the domain boundary will decrease the leading-order term µ0 in the asymptotic expansion of the principal eigenvalue λ in (5.2.27) of Principal Result 5.3. Moreover, for a convex domain with piecewise smooth boundary, Qualitative Result I together with Principal Result 5.4 shows that µ0 will be reduced the most by the movement of an interior patch to a non-smooth boundary point with the smallest corner contact angle. 140 5.3. Effect of Habitat Fragmentation and Location on Species Persistence Since this patch was chosen arbitrarily, it is clear that µ0 is minimized by the placement of all interior patches to the boundary of the domain. This feature is encapsulated in the following qualitative result: Qualitative Result I: The movement of either a single favourable or unfavourable habitat to the boundary of the domain is advantageous for the persistence of the species. Next, the effect of fragmentation on species persistence is considered. More specif- ically, the effect of splitting the ith patch, of radius ερi and growth rate mi, into two distinct patches, one with radius ερj and growth rate mj, and the other with radius ερk and growth rate mk. The condition miρ 2 i = mjρ 2 j +mkρ 2 k is imposed to satisfy the constraint (5.3.1). To isolate the role of fragmentation, the assumption αi = αj = αk is made so that either an interior patch is split into two interior patches, or a boundary patch into two boundary patches, with each boundary patch centred at either a smooth point of ∂Ω or at a corner point of ∂Ω with the same contact angle. This action leads to the following qualitative result: Qualitative Result II: The fragmentation of one favourable interior habitat into two separate favourable interior habitats is not advantageous for species persistence. Simi- larly, the fragmentation of a favourable boundary habitat into two favourable boundary habitats with each either centred at either a smooth point of ∂Ω, or at a corner point of ∂Ω with the same contact angle, is not advantageous. Finally, the fragmentation of an unfavourable habitat into two separate unfavourable habitats increases the persistence threshold λ. First, consider the case where one favourable habitat is broken into two smaller favourable habitats. Then, mi > 0, mj > 0, and mk > 0. For the original patch distribution, it follows from § 5.2 that Bold(ζ) = 0 has a positive root ζ = µold0 on 0 < ζ < µoldm ≡ 2/(mJρ2J), wheremJρ2J was defined in (5.2.30). Clearly the first vertical asymptote for Bnew(ζ) cannot be smaller than that of Bold(ζ) under this fragmentation, therefore it follows from § 5.2 that Bnew(ζ) = 0 has a positive root ζ = µnew0 on 0 < ζ < µnewm with µ new m ≥ µoldm . From (5.2.28), the change in B(ζ) induced by this fragmentation action is calculated under the constraint miρ 2 i = mjρ 2 j +mkρ 2 k: Bnew(ζ)− Bold(ζ) = παimjρ 2 j (2− ζmjρ2j) + παimkρ 2 k (2− ζmkρ2k) − παimiρ 2 i (2− ζmiρ2i ) = −παiζ(mjρ2jmkρ2k) [ (2− ζmjρ2j ) + (2− ζmkρ2k) ] (2− ζmiρ2i )(2− ζmjρ2j )(2− ζmkρ2k) . (5.3.13) Hence, from (5.3.13), it is clear that Bnew(ζ) < Bold(ζ) on 0 < ζ < µoldm ≡ 2/(mJρ2J). Since, in addition µnewm ≥ µoldm , it follows from Case II of Lemma 4.4 that µnew0 > µold0 . 141 5.3. Effect of Habitat Fragmentation and Location on Species Persistence This proves the first two statements of Qualitative Result II. To prove the final statement of this result, suppose that an unfavourable habitat is broken into two smaller unfavourable habitats, so that mi < 0, mj < 0, and mk < 0. For this situation, the first vertical asymptotes of Bold(ζ) and Bnew(ζ) are the same, and (5.3.13) again shows that Bnew(ζ) < Bold(ζ) on 0 < ζ < µoldm ≡ 2/mJρ2J . By Case II of Lemma 4.4, we conclude that µnew0 > µ old 0 , which proves the last statement of Qualitative Result II. The combination of Qualitative Results I and II show that, given some fixed amount of favourable resources to distribute, the optimal strategy is to clump them all together at a point on the boundary of the domain, and more specifically at the corner point of the boundary (if any are present) with the smallest contact angle less than π degrees. This strategy will ensure that the value of µ0, and consequently the leading-order term for λ, is as small as possible, thereby maximizing the range of diffusivities D in (1.2.1) for the persistence of the species. Our final qualitative result addresses whether or not it is advantageous to fragment a single interior favourable habitat into a smaller interior favourable habit together with a favourable boundary habitat. To study this situation, consider the constraint miρ 2 i = mjρ 2 j + αk 2 mkρ 2 k , (5.3.14) with αi = αj = 2, and αk < 2. The subscript i represents the original interior habitat, whereas j and k represent the new smaller interior habitat and new boundary habitat, respectively. It is not clear a priori whether or not this action is advantageous, given that fragmentation of a favourable interior habitat into two favourable interior habitats increases the persistence threshold λ, but the relocation of a favourable interior habitat to the boundary decreases λ. A sufficient condition to treat this case, together with two additional related results, are summarized as follows: Qualitative Result III: The fragmentation of one favourable interior habitat into a new smaller interior favourable habitat together with a favourable boundary habitat, is advantageous for species persistence when the boundary habitat is sufficiently strong in the sense that mkρ 2 k > 4 2− αk mjρ 2 j > 0 . (5.3.15) Such a fragmentation of a favourable interior habitat is not advantageous when the new boundary habitat is too weak in the sense that 0 < mkρ 2 k < mjρ 2 j . (5.3.16) Finally, the clumping of a favourable boundary habitat and an unfavourable interior 142 5.3. Effect of Habitat Fragmentation and Location on Species Persistence habitat into one single interior habitat is not advantageous for species persistence when the resulting interior habitat is still unfavourable. To prove this result, first impose the constraint (5.3.14), and then calculate from (5.2.28) that Bnew(ζ)− Bold(ζ) = 2πmjρ 2 j (2− ζmjρ2j) + παkmkρ 2 k (2− ζmkρ2k) − 2πmiρ 2 i (2− ζmiρ2i ) , = παkζβk (2− ζβi)(2 − ζβj)(2− ζβk)   (2− αk)βk − 4βj +ζβj ( βj + αk 2 βk )   , (5.3.17a) = παkζβk (2− ζβi)(2 − ζβj)(2− ζβk)   2(βk − βj)− βj(2− ζβj) −αkβk 2 (2− ζβj)   , (5.3.17b) where βi ≡ miρ2i , βj ≡ mjρ2j , and βk ≡ mkρ2k. There are three parameter ranges of interest, corresponding to the three statements in Qualitative Result III. First suppose that βi > 0 and βk > 4 2−αk βj > 0. Then, from (5.3.14), it follows that βi > βj , and βi < (2− αk) 4 βk + αk 2 βk = βk − 1 2 ( 1− αk 2 ) βk , so that βi < βk since 0 < αk < 2. It then readily follows that the first vertical asymptote µnewm and µ old m for Bnew(ζ) and Bold(ζ), respectively, must satisfy µnewm ≤ µoldm . Furthermore, it follows from (5.3.17a) that Bnew(ζ) > Bold(ζ) on 0 < ζ < µnewm . Consequently, Case I of Lemma 5.7 ensures that µnew0 < µ old 0 . This establishes the first statement of Qualitative Result III. Second, suppose that βi > 0 and βj > βk > 0. Then, from (5.3.14), it follows that βi > βj , and βi > βk + αkβk/2 > βk since 0 < αk < 2. The condition that βi > βj and βi > βk ensures that the first vertical asymptotes of Bnew(ζ) and Bold(ζ) must satisfy µnewm ≥ µoldm . Furthermore, it follows from (5.3.17b) that Bnew(ζ) < Bold(ζ) on 0 < ζ < µoldm . Consequently, Case II of Lemma 5.7 yields that µ old 0 < µ new 0 . This establishes the second statement of Qualitative Result III. Finally, suppose that βj < 0, βk > 0, and βi = βj +αkβk/2 < 0. Then, since βi < 0, it follows that the first vertical asymptote µoldm for Bold(ζ) cannot occur from the ith patch. The condition βk > 0 then ensures that µ new m ≤ µoldm , where µnewm is the vertical asymptote of Bnew(ζ). Furthermore, it follows from (5.3.17a) that Bnew(ζ) > Bold(ζ) on 0 < ζ < µnewm . Consequently, Case I of Lemma 5.7 establishes that µ old 0 < µ new 0 , which proves the final statement of Qualitative Result III. As a remark, an interpretation of the first statement of Qualitative Result III is 143 5.3. Effect of Habitat Fragmentation and Location on Species Persistence provided in terms of the areas of the patches for the special case where mj = mk = 1. Then, from (5.3.15) it follows that the fragmentation of a favourable interior habitat is advantageous when the area ε2Ak ≡ πε2ρ2k/2 of a new favourable habitat centred at a smooth point of the boundary is at least twice as large as the area ε2Aj ≡ πε2ρ2j of the new smaller favourable interior habitat. If the new boundary habitat is located at a π/2 corner of the domain, for which αk = 1/2, then a sufficient condition for this fragmentation to be advantageous is when the area ratio satisfies Ak/Aj = ρ 2 k/(4ρ 2 j ) > 2/3. The following examples are provided to illustrate Qualitative Results I–III. Example I: (The Unit Disk): Let Ω be the unit disk, a domain for which the Neu- mann Green’s function and its regular part are known explicitly ( c.f. Appendix B of [31] ) to be G(x;x0) = −1 2π log |x− x0|+R(x;x0) , (5.3.18a) R(x;x0) = −1 2π ( log ∣∣∣∣x|x0| − x0|x0| ∣∣∣∣− 12(|x|2 + |x0|2) + 34 ) , (5.3.18b) We will compare the two-term asymptotic formula for λ in (5.2.27) of Principal Result 5.3 for three different arrangements of favourable resources inside Ω with mb = 2 and mj = 1 for j = 1, . . . , n. For each of the three arrangements below, the constraint (5.3.1) has fixed value ∫ Ωmdx = −π. In the first example, favourable resources are clumped into one interior patch centred at the origin of radius ε. Substituting m+ = 1, mb = 2, |Ω| = π, and R(0; 0) = −3/(8π) from (5.3.18b), into (5.1.26a) yields that λ ∼ ν + ν2/2 , (interior patch) , (5.3.19) where ν = −1/ log ε. Next, consider the optimal case where the favourable resources are all concentrated at a patch of radius √ 2ε that is centred on the boundary of the unit disk. Since Ω is a disk, any such boundary point x0 yields the minimum value of λ. Substituting m+ = 1, mb = 2, |Ω| = π, α0 = 1, ρ0 = √ 2, and Rs(x0;x0) = 1/(8π) from (5.3.18b), into (5.1.46a) gives that λ ∼ ν 2 − ν 2 2 ( 3 8 − log √ 2 ) , (boundary patch) . (5.3.20) The next example supposes that n favourable patches of a common smaller radius ε/ √ n have centres at the equally spaced points xj = re 2piij/n on a ring of radius r < 1, where i = √−1. In this case, set mb = 2, |Ω| = π, mj = 1, ρj = 1/ √ n, and αj = 2 for 144 5.3. Effect of Habitat Fragmentation and Location on Species Persistence 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.100.080.060.040.020.00 λ ǫ (a) λ versus ε 1.6 1.4 1.2 1.0 0.8 0.6 0.4 1.00.80.60.40.20.0 µ0 ρ1 • • (b) µ0 versus ρ1 Figure 5.4: Example 1: Choose mb = 2, and mj = 1 for j = 1, . . . , n, in the unit disk with ∫ Ωmdx = −π. Left figure: λ versus ε for three different cases: a single boundary patch (5.3.20) (heavy solid curve); a single interior patch centred at the origin (5.3.19) (solid curve); four small patches equally spaced on a ring of radius r = 0.5 (5.3.21) (dashed curve). The boundary patch gives the smallest λ, followed by the non-fragmented interior patch solution. Right figure: the leading order coefficient µ0 versus ρ1 from (5.3.22b) for the partial fragmentation of an interior patch of radius ε into a smaller interior patch of radius ερ1 together with a boundary patch of radius ερ0, while maintaining ∫ Ωmdx = −π. The bullets indicate the bounds from (5.3.15) and (5.3.16) of Qualitative Result III. Fragmentation is advantageous only when ρ1 < √ 2/5. j = 1, . . . , n, in (5.2.31) for µ0 and (5.2.27) for λ. In this way, µ0 = n, and the two-term expansion of λ (5.2.27) reduces to λ ∼ nν − nν2 ( qn(r) + 1 2 log n+ 1 4 ) , qn(r) ≡ 2π n2 pn(r) , (5.3.21a) where pn(r) ≡ neTG(N)e. Here e = (1, . . . , 1)T , and G(N) is the n × n Neumann Green matrix with matrix elements (G(N)) ij = G(xi;xj) for i 6= j and (G(N)) jj = R(xj ;xj), where G(xi;xj) and R(xj ;xj) are the Neumann Green’s function of (5.1.19), given explicitly for the unit disk in (5.3.18). For n equally spaced patch centres on a ring of radius r < 1, pn(r) can be calculated explicitly, and is given in Proposition 4.3 of [57]. In this way, a direct calculation of qn(r) in (5.3.21a) gives qn(r) = r 2 − 3 4 − 1 n log ( nrn−1 )− 1 n log ( 1− r2n) . (5.3.21b) In Fig. 5.4(a), the three different two-term expansions for λ versus ε, given in (5.3.19), (5.3.20), and (5.3.21) with n = 4 and ring radius r = 1/2, representing the three different spatial arrangements of favourable resources. In agreement with predictions made in Qualitative Results I and II, the best choice is to concentrate resources on the boundary of the domain, while clumping resources at the centre of the domain provides a better 145 5.3. Effect of Habitat Fragmentation and Location on Species Persistence alternative than fragmenting the favourable resources into four separate patches on a ring. The next example is an illustration of Qualitative Result III. Consider fragmenting a single interior patch solution of radius ε centred at the origin into a boundary patch of radius ερ0 and a smaller interior patch of radius ερ1, while maintaining ∫ Ωmdx = −π. Thus, ρ0 and ρ1, with 0 < ρ1 < 1, must satisfy the constraint 1 = ρ21 + 1 2 ρ20 . (5.3.22a) As remarked following (5.2.31), for a two-patch problem (5.2.28) reduces to a quadratic equation. In this case, that quadratic equation in µ0 is µ20ρ 2 1 ( 1− ρ21 ) + µ0 ( −2 + 5 2 ρ21 − 3 2 ρ41 ) + 1 = 0 . (5.3.22b) Notice that µ0 = 1 when ρ1 = 1, and µ0 = 1/2 when ρ1 = 0, as expected. A plot of the smallest root to (5.3.22b) versus ρ1 is shown in Fig. 5.4(b). The bound (5.3.16) in Qualitative Result III states that the partial fragmentation of the interior patch into a boundary patch is undesirable when ρ1 > ρ0, which yields ρ1 > √ 2/3 from (5.3.22a). Alternatively, (5.3.15) together with (5.3.22a) shows that such a fragmentation is ad- vantageous when ρ1 < 1/ √ 3. These two bounds are shown by the bullets in Fig. 5.4(b). For this simple two-patch case, we can readily show from the exact result (5.3.22b) that µ0 = 1 when ρ1 = √ 2/5, or equivalently ρ0 = √ 6/5. Thus, fragmentation is advantageous when ρ1 < √ 2/5, or equivalently ρ0 > √ 6/5. The final example illustrates Qualitative Result III for the case where the unit disk has one pre-existing favourable interior patch of radius ε and growth rate m+ = 1, together with one pre-existing unfavourable interior patch of radius ε and growth rate m− = −1. An additional favourable resource of area ε2A0, if separated from the other two patches, is introduced with local growth rate m0 = 1. The bulk decay rate is chosen to be mb = 3. Three different possible options for using this additional favourable resource are now explored, subject to the constraint that ∫ Ωmdx = −3π +A0 remains fixed. If the additional favourable resources are concentrated at a smooth point on the boundary, then from (5.2.28) µ0 satisfies − 3 + 2 ( 1 2− µ0 − 1 2 + µ0 ) + A0/π 1− µ0A0/π = 0 . (5.3.23a) Alternatively, if the additional favourable resource is used to strengthen the pre-existing 146 5.3. Effect of Habitat Fragmentation and Location on Species Persistence favourable interior patch, then from (5.2.28) µ0 satisfies − 3 + 2ρ 2 + 2− ρ2+µ0 − 2 2 + µ0 = 0 , ρ2+ = 1 +A0/π . (5.3.23b) Finally, if the additional favourable resource is used to diminish the strength of the unfavourable pre-existing interior patch, then µ0 satisfies − 3 + 2 2− µ0 + m− 2−m−µ0 = 0 , m− = −1 +A0/π . (5.3.23c) In Fig. 5.3.2, the three curves for µ0 obtained from (5.3.23a)–(5.3.23c) are plotted against A0/π. A zoom of Fig. 5.5(a) for a subrange of A0/π is shown in Fig. 5.5(b). From the plots we conclude that inserting a favourable boundary patch is preferable only when it has a sufficiently large size, and that if one only has a limited amount of an additional favourable resource, it is preferable to re-enforce the pre-existing favourable habitat. In addition, Fig. 5.5(b) shows that it is not optimal for any range of A0/π to use the additional favourable resource to mitigate the effect of the unfavourable interior patch. 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 2.52.01.51.00.50.0 µ0 A0/pi (a) µ0 versus A0/pi 0.5 0.4 0.3 0.2 0.1 0.0 2.42.22.01.81.61.41.2 µ0 A0/pi (b) µ0 versus A0/pi Figure 5.5: Example 1: Choose mb = 3 and consider a pre-existing patch distribution of one favourable interior patch of local growth rate m+ = 1 and radius ε and an unfavourable interior patch of local growth rate m− = −1 and radius ε. Assume an additional favourable resource of local growth rate m0 = 1 that can occupy an area ε 2A0 if it separated from the two pre- existing interior patches. Plotted is µ0 versus A0/π when the additional resource is on the domain boundary (5.3.23a) (heavy solid curve), when it is used to re-enforce the existing favourable interior patch (5.3.23b) (solid curve), and when it is used to mitigate the effect of the unfavourable interior patch (5.3.23c) (dotted curve). 147 5.3. Effect of Habitat Fragmentation and Location on Species Persistence 5.3.3 Optimization at Second Order The minimization of λ in (5.2.27) is typically accomplished from the optimization of the coefficient µ0 of the leading-order O(ν) term in its asymptotic expansion. However, in certain degenerate cases, such as if a fixed distribution of interior patches is already present in the domain, the problem of optimizing the persistence threshold λ can require a careful examination of the coefficient µ1 of the O(ν2) term in the asymptotic expansion of λ in (5.2.27). The coefficient µ1 has an explicit dependence on the patch locations and accounts for interaction effects between the patches. An optimization problem of this type occurs in choosing the best location to place an additional favourable resource in a square domain. If this resource is sufficiently strong, then to minimize µ0 it should be centred on the boundary of the square at a π/2 corner, and should not be used to strengthen a pre-existing favourable interior patch. In the situation where no other interior patches are present, each of the four corners of the square offers an equally good location to concentrate resources. However, if a distribution of fixed patches is already present in the domain, the best choice of corner to place an additional favourable patch is not clear a priori, and will depend on the spatial configuration of the fixed pre-existing patch distribution. In this case, the information required to make the optimal choice is provided by µ1, which takes into account the interaction between the patches. To formulate this restricted optimization problem, define xj for j = 1, . . . , n to be a fixed pre-existing configuration of n circular patches in the interior of the domain whose local growth rates are mj for j = 1, . . . , n respectively, where mj is either positive (favourable habitat) or negative (unfavourable habit). Now consider the introduction of a single new favourable habitat, and assume that µ0 is smallest when this additional habitat is located on the boundary of the domain. We then consider the problem of determining the optimal boundary location, x0, of the centre of the additional circular patch whose radius is ερ0, local growth rate m0 > 0 and angle πα0. Earlier analysis showed that to optimize µ0, x0 should be centred at a boundary point with the smallest contact angle πα0. In degenerate situations where this point is not uniquely determined, the coefficient of theO(ν2) term in (5.2.27) must be optimized. To do so, label xn+1 = x0 and block the (n + 1) × (n + 1) matrices in (5.2.27) into an n × n block, labeled by Gm and P, representing the fixed patch distribution, and a term p(x0) representing the interaction of the fixed patch distribution with the additional favourable resource. This determines µ1 in (5.2.27) as µ1 = µ0 ( −1 4 + κT (P − πGm)κ+ κ20 log ρ0 − πp(x0) κTκ+ κ20 ) . (5.3.24a) In terms of the fixed distribution of patches, κ = (κ1, . . . , κn) T , where κj for j = 1, . . . , n 148 5.3. Effect of Habitat Fragmentation and Location on Species Persistence is defined in (5.2.22), while Gm and P are the n × n matrices as defined in (5.2.23). The scalar p(x0) in (5.3.24a), representing the interaction of the additional favourable boundary patch, centred at x0, with the fixed patch distribution is given by p(x0) = α0κ 2 0Rm(x0;x0) + 2 n∑ j=1 √ α0αjκ0κjGm(xj ;x0) , κ0 ≡ √ α0m0ρ 2 0 2− µ0m0ρ20 , (5.3.24b) where αj = 2 for j = 1, . . . , n. From (5.2.28), the leading-order coefficient µ0 in the asymptotic expansion of λ is the smallest positive root of −mb|Ω|+π √ α0κ0+π n∑ j=1 √ αjκj = 0 , κj ≡ √ αjmjρ 2 j 2− µ0mjρ2j , j = 1, . . . , n . (5.3.25) The minimization of the persistence threshold λ corresponds to determining the location of the maximum of p(x0) for x0 ∈ ∂Ω. The problem of maximizing p(x0) is now illustrated for two specific examples. Example 2: Pre-Existing Patch Distribution (Unit Disk): Let Ω be the unit disc and x0 ∈ ∂Ω, for which α0 = 1. Since x0 ∈ ∂Ω, then Gm(xj ;x0) = Gs(xj ;x0) and Rm(x0;x0) = Rs(x0;x0) are the surface Neumann Green’s function and its regular part given explicitly in (5.3.18) after setting |x0| = 1. Now consider placing the centres xj for j = 1, . . . , n of the fixed patches on a ring of radius r so that xj = r exp (2πij/n) and αj = 2 for j = 1, . . . , n, with 0 < r < 1. Then, from (5.3.24b) and (5.3.18), and with α0 = 1, we obtain p(x0) = κ20 8π + 2κ0 n∑ j=1 √ αjκj [ r2 4π − 1 8π − 1 2π log |xj − x0|2 ] , = κ20 8π + κ0 2π ( r2 − 1 2 ) n∑ j=1 √ αjκj − √ 2κ0 π n∑ j=1 κj log |xj − x0|2 , = κ20 8π + κ0 2π ( r2 − 1 2 ) (mb − κ0)− √ 2κ0 π n∑ j=1 κj log |xj − x0|2 , (5.3.26) where in the last equality the identity ∑n j=1 √ αjκj = mb − κ0 from (5.3.25) has been used. Finally, writing x0 = e iθ0 and calculating the logarithmic interaction term in (5.3.26) yields that p = p(θ0), where p(θ0) = κ20 8π + κ0 2π ( r2 − 1 2 ) (mb − κ0)− √ 2κ0 π n∑ j=1 κj log ( r2 + 1− 2r cos ( θ0 − 2πj n )) . (5.3.27) The location of the maximum value of p(θ0) in (5.3.27) is now determined, which cor- 149 5.3. Effect of Habitat Fragmentation and Location on Species Persistence responds to the optimum location to insert the additional favourable resource on the boundary of the unit disk. 7 6 5 4 3 2 1 0 1.00.80.60.40.20.0 µ0 A0/pi Figure 5.6: Example 2: Choosemb = 3 and centre five patches each of radius ε/ √ 5 equidistantly on a ring of radius r = 1/2 in the unit disk with mj = 1 for j = 1, . . . , 5. Plot of µ0, defined as the root of (5.2.28), versus A0/π for an additional patch of area ε2A0 located on the boundary of the domain (heavy solid curve) or used to re-enforce any one of the interior patches (solid curve). The boundary patch is preferable when A0/π = 1/2, which corresponds to the boundary patch radius ρ0 = 1. The bound (5.3.15) states that the boundary patch is favourable when its radius satisfies ρ0 > 2/ √ 5 (or A0/π > 2/5), while from (5.3.16) the boundary patch is not favourable when ρ0 < 1/ √ 5 (or A0/π < 1/10). By supposing that mj = mc for j = 1, . . . , n, so that κj = κc for j = 1, . . . , n, i.e. asserting that all patches on the ring have the same strength, (5.3.27) simplifies to p(θ0) = κ20 8π + κ0 2π ( r2 − 1 2 ) (mb − κ0)− √ 2κ0κc π χ(θ0) , χ(θ0) ≡ n∑ j=1 log [ r2 + 1− 2r cos ( θ0 − 2πj n )] . (5.3.28) The function χ(θ0) is reduced further by the following steps χ(θ0) = n∑ j=1 log [( r − cos ( θ0 − 2πj n ))2 + sin2 ( θ0 − 2πj n )] = 2 log ( n∏ j=1 |r − zj| ) = 2 log |rn − einθ0 | = log [(rn − cos(nθ0))2 + sin2(nθ0)] , (5.3.29) where zj ≡ ei(θ0−2pij/n). In obtaining the second to last equality in (5.3.29), the fact that zj are the roots of r n− einθ0 = 0 has been used. Upon differentiating (5.3.29) with respect to θ0, it readily follows that the critical points of χ(θ0), and therefore p(θ0), 150 5.3. Effect of Habitat Fragmentation and Location on Species Persistence satisfy sin(nθ0) = 0, which admits the 2n solutions θ0 = 2πj n , θ0 = π(2j − 1) n , j = 1, . . . , n , (5.3.30) on the interval 0 < θ0 ≤ 2π. When κc > 0 in (5.3.28), then θ0 = 2πj/n for j = 1, . . . , n, clearly correspond to maxima of p(θ0), while the remaining critical points in (5.3.30) are minima of p(θ0). This result shows that when κc > 0, for which the ring is composed of n equally distributed favourable patches, the optimal boundary locations for the one additional favourable patch centred at x0 is at the shortest distance to any of the n favourable habits on the ring. This result for p(θ0) is illustrated by the heavy solid curve of Fig. 5.7(a) for n = 5 pre-existing patches for the parameter set mb = 3, m0 = 1, ρ0 = 1, and with mj = 1 and ρj = 1/ √ 5 for j = 1, . . . , 5. For this parameter set, where the favourable boundary patch is sufficiently strong in the sense of (5.3.15) of Qualitative Result III, µ0 is indeed minimized when the favourable resource is concentrated on the boundary of the domain, rather than being used to re-enforce an interior favourable habitat. In Fig. 5.6, µ0 is plotted against A0/π, where ε 2A0 is the area of the additional favourable habitat for the case where the habitat is located on the boundary, and for the case when it is used to re-enforce one of the pre-existing favourable interior habitats. From this plot, we observe that a boundary habitat with ρ0 = 1 and A0/π = 1/2 provides µ0 = 1.455, which is the smaller of the two values for µ0. Alternatively, when κc < 0, only the locations θ0 = π(2j − 1)/n for j = 1, . . . , n correspond to maxima of p(x0). For this case, where the ring is composed of n equally spaced unfavourable habitats, the optimal boundary locations for the one additional favourable patch centred at x0 is such that it maximizes the distance to any of the n unfavourable habitats on the ring. This result for p(θ0) is illustrated by the solid curve in Fig. 5.7(a) for n = 5 pre-existing patches for the parameter set mb = 3, m0 = 1, ρ0 = 1, and with mj = −1 and ρj = 1/ √ 5 for j = 1, . . . , 5. For this parameter set µ0 = 1.740. 151 5.3. Effect of Habitat Fragmentation and Location on Species Persistence 0.8 0.6 0.4 0.2 0.0 6543210 p(θ0) θ0 (a) p(θ0) versus θ0 1.2 1.0 0.8 0.6 0.4 0.2 0.0 6543210 p(θ0) θ0 (b) p(θ0) versus θ0 Figure 5.7: Example 2: Choosemb = 3 and centre five patches each of radius ε/ √ 5 equidistantly on a ring of radius r = 1/2 in the unit disk. Insert a favourable boundary patch of radius ρ0 = 1 and growth rate m0 = 1 at x0 = e iθ0 on the boundary of the unit disk. Left figure: p(θ0) versus θ0 from (5.3.27) for favourable interior patches (heavy solid curve) with mj = 1 for j = 1, . . . , 5, and for unfavourable interior patches (solid curve) with mj = −1 for j = 1, . . . , 5. Right figure: p(θ0) versus θ0 for four unfavourable interior patches with mj = −1 for j = 1, . . . , 4 and one favourable interior patch at x1 = (r, 0) with m5 = 1. A further example considers the case of n patches with a common radius ερc but with mj = −mc for j = 1, . . . , n − 1, and mn > 0, where mc > 0. Therefore, there are n − 1 unfavourable habitats on the ring, with the only favourable habitat on the ring being centred at xn = (r, 0). For this case, p(θ0) is given by (5.3.27) provided that we replace κj in (5.3.27) with κj ≡ − √ 2mcρ 2 c 2 + µ0mcρ2c , j = 1, . . . , n− 1 , κn ≡ mnρ 2 c 2− µ0mnρ2c . (5.3.31) A simple calculation from (5.3.27) shows that the maximum of p(θ0) occurs at θ0 = 0. Therefore, the best location for the favourable boundary habitat is to insert it as close as possible to the only favourable interior habitat on the ring, which effectively decreases the effect of fragmentation. This result for p(θ0) is illustrated by the dashed curve in Fig. 5.7(b) for n = 5 pre-existing patches for the parameter set mb = 3, m0 = 1, ρ0 = 1, mj = −1 and ρj = 1/ √ 5 for j = 1, . . . , 4, and m5 = 1 with ρ5 = 1/ √ 5. For this parameter set µ0 = 1.709. The figure shows that p(θ0) is minimized when θ0 = π, corresponding to the location on ∂Ω furthest from the only favourable interior habitat. Example 3: Pre-Existing Patch Distribution (The Unit Square) The final ex- ample considers the problem of optimizing the location of one additional favourable resource in the unit square domain Ω = [0, 1] × [0, 1] given a certain distribution of pre-existing patches. This optimization problem is somewhat simpler than the previous example for the unit disk, since if the patch is sufficiently strong, the optimization of 152 5.3. Effect of Habitat Fragmentation and Location on Species Persistence µ0 requires that the additional resource be located at the corner x0 of the square that maximizes p(x0) in (5.3.24b). For the unit square, explicit analytical formulae for the Neumann Green’s function and its regular part, required to optimize p(x0), are obtained in §3.2 of [62]. The calculations leading to these formulae are lengthy and so only the results are stated. The Neumann Green’s function with an interior singularity is given by G(x;x0) = − 1 2π log |x− x0|+R(x;x0) , (5.3.32a) where the regular part R(x;x0) is given explicitly by R(x;x0) = − 1 2π ∞∑ n=0 log(|1− qnz+,+||1− qnz+,−||1− qnz−,+||1 − qnζ+,+| × |1− qnζ+,−||1− qnζ−,+||1− qnζ−,−|) − 1 2π log |1− z−,−| |r−,−| +H(x (1), x (1) 0 )− 1 2π ∞∑ n=1 log |1− qnz−,−| , (5.3.32b) and x = (x(1), x(2)) and x0 = (x (1) 0 , x (2) 0 ). Here the eight complex constants z±,± and ζ±,± are defined in terms of additional complex constants r±,±, ρ±,± by z±,± ≡ epir±,± , ζ±,± ≡ epiρ±,± , q ≡ e−2pi < 1 , (5.3.33a) r+,± ≡ −|x(1) + x(1)0 |+ i(x(2) ± x(2)0 ) , r−,± ≡ −|x(1) − x(1)0 |+ i(x(2) ± x(2)0 ) , (5.3.33b) ρ+,± ≡ |x(1) + x(1)0 | − 2 + i(x(2) ± x(2)0 ) , ρ−,± ≡ |x(1) − x(1)0 | − 2 + i(x(2) ± x(2)0 ) . (5.3.33c) In (5.3.32) and (5.3.33), |ω| is the modulus of the complex number ω. In (5.3.32b), H(x(1), x (1) 0 ) is defined by H(x(1), x (1) 0 ) ≡ 1 12 [ h(x(1) − x(1)0 ) + h(x(1)1 + x(1)0 ) ] , h(θ) ≡ 2−6|θ|+3θ2 . (5.3.34) The self-interaction term R(x0;x0), required in (5.1.19b) is obtained by setting x = x0 in (5.3.32b). Let x0 be the location of a patch of favourable resource with radius ε and local growth rate m0 = 1 and assume that there is a configuration of four pre-existing patches in Ω centred at x1 = (1/4, 1/4), x2 = (1/4, 3/4), x3 = (3/4, 1/4), and x4 = (3/4, 3/4). Let each patch have a common radius ε/2 (i.e. ρj = 1/2 for j = 1, . . . , 4) with local growth rate m1 = m2 = m3 = −1, m4 = 1, and the background decay rate mb = 3. For this parameter set, where the favourable boundary patch is sufficiently strong in the 153 5.4. Conclusions sense of (5.3.15) of Qualitative Result III, µ0 is minimized when the favourable resource is concentrated at one of the four corners on the square, rather than being used to re-enforce the only favourable interior habitat. By determining the root µ0 of (5.2.28) numerically, the value µ0 = 1.605 is obtained when the additional favourable resource is at a corner of the square, and µ0 = 2.681 when the additional favourable resource is used to strengthen the favourable resource at x4. Therefore, µ0 is smallest when x0 is at a corner of the square. Then, by varying x0 over the four corners of the square, the following numerical results for p(x0) are obtained from (5.3.24b): x0 = (0, 0) p(x0) = −0.8522 ; x0 = (1, 1) p(x0) = −0.2100 . x0 = (1, 0) or x0 = (0, 1) p(x0) = −0.7163 ; The largest value for p(x0) occurs when x0 = (1, 1). Therefore, these results show that the persistence threshold λ is smallest when the additional favourable habitat is positioned at the corner of the square that is closest to the only favourable interior habitat. This action effectively decreases the effect of fragmentation. In cases where µ0 is lowest positive value at a discrete set of points, as in the previous example, optimization of the O(ν2) terms in λ can certainly be achieved by calculating µ1 at each of these discrete points and sorting for the smallest. This method does rely on the availability of values for R(x;x0), which may in general be calculated numerically as in [62]. In the previous example, the optimal choice was that which minimized the euclidean distance between the one existing favourable resource and the favourable additional resource. Heuristically, this action can be thought of as effectively decreasing the the fragmentation of the resource configuration. For more complex configurations however, it is not clear if a rule of thumb can be established, as consideration of a more complicated fragmentation metric may be required. 5.4 Conclusions For a specific but fairly general class of functions m(x), a definitive strategy for the optimization of the principal eigenvalue of (5.0.1) has been established. For a class of m(x) consisting of a background decay rate plus localized patches of either highly favourable or unfavourable rates, a two term expansion of the persistence threshold was developed in (5.2.27). Interestingly, a large quantity of information regarding the optimal configuration of patches is contained in the leading order term even although it has no explicit depen- dence on the path centres. In certain degenerate cases, the second order term must be considered to find the optimal configuration of resources. 154 5.4. Conclusions By directly optimizing the persistence threshold λ, Qualitative Results I–III were obtained in § 5.3.2. In particular it was shown that moving a favourable patch from an interior location to a boundary location is always advantageous to species persis- tence. If the boundary of the domain has corners, then the corner with the smallest angle produces the lowest persistence threshold. It was also shown that splitting one favourable interior patch into two favourable interior patches always increased the per- sistence threshold and in this sense, fragmentation can be thought of a deleterious to the well-being of a species evolving in Ω. The division of a single interior patch into a boundary patch and an interior patch is only advantageous provided the boundary patch is sufficiently strong. The findings reported here agree well with known optimization results for the 1D strip ( c.f. [60, 46] ) and numerical results in higher dimensions (c.f. [42]). This work forms the basis of paper [31], titled An Asymptotic Analysis of the Persis- tence Threshold for the Diffusive Logistic Model in Spatial Environments with Localized Patches to appear in Discrete and Continuous Dynamical Systems Series B. 155 Chapter 6 Discussion In this section, I will provide an overview of the results of the two problems studied and place these results in context with the published literature on models of MEMS devices and Mathematical Ecology. There are several areas where open problems remain for which further investigation is warranted and in relation to this work, several of these will be discussed. 6.1 Micro-Electro Mechanical Systems The motivation for our mathematical study of the models of Micro-Electro Mechanical Systems is two fold. The first motivation is the study of the pull-in phenomena which manifests itself mathematically as a saddle-node bifurcation in a non-linear eigenvalue problem. The pull-in instability is crucial to the effective design of MEMS devices and so characterizing it mathematically is an important and interesting problem. Previous work in the mathematical study of the pull-in stability has furnished the area with many important rigorous results, particularly those pertaining to the existence, stability and regularity of an extremal solution at the end of the minimal solution branch for (1.1.3) [16]. While our work lacks a rigorous basing, indeed many of our results can be thought of as predictions, its advantage lies in its explicit characterization of the solution structure. Numerical evidence can be useful to bolster the veracity of our predictions, but a rigorous analysis would certainly complement our findings well. In Chapter 3, the location of this fold point was determined explicitly by means of matched asymptotic expansions for three particular models, the beam problem (1.1.4), the fringing fields problem (1.1.5) and the annulus problem (1.1.6). A comparison with full numerics shows that the location of the fold point is accurately predicted by asymp- totic calculations for a large range of physical parameters. Accurate determination of the fold point located at the end of the minimal branch is crucial in the design of MEMS devices as typically such devices are operated very close to this threshold. The asymp- totic formulae derived in result (3.3.19) may therefore be useful to MEMS practitioners who rely on accurate determination of this value. By including more physical effects in the modeling, this may add refinement to the current practice of MEMS design. The second motivation for the mathematical study of MEMS models concerns the 156 6.1. Micro-Electro Mechanical Systems nature of singular solutions as ||u||∞ → 1 which are dominated by the λ/(1 + u)2 nonlinearity ubiquitous in equations of MEMS. In this regime, several rigorous results pertaining to solution multiplicity have been established (c.f. [20, 13, 9]), in particular, it has been established that (1.1.2) exhibits an infinite number of solutions. Building on this understanding, the asymptotic techniques developed in this thesis allows for these singular solutions to be constructed explicitly for certain domains. This explicit characterization allows for very particular predictions to be made on the nature of the solutions. One common feature in the analysis of these singular solutions is the definition ||u||∞ = 1− ε, i.e. the maximum absolute deflection of the plate is specified to be some small distance from 1. This allows the resolution, by means of asymptotic methods, of the λ/(1 + u)2 nonlinearity near the point of maximum absolute deflection and so facilitates the construction of singular solutions as ε → 0+. It is shown in § 4.1 that solutions to (1.1.2) for the unit disc exhibit an infinite number of fold points centred on a determined value of λ = λ∗ > 0 as ||u||∞ → 1. The locations of these fold points are determined explicitly in the analysis and are observed to be exponentially close to u = −1. Predictions are shown to agree very well with numerical calculations (see Fig. 4.3). While the fringing fields, beam and annulus perturbations induce a quantitative change on the location of the pull-in instability, these same perturbations are observed to destroy the infinite fold point structure and the bifurcation diagrams of the three problems are left with a finite number of fold points followed by limiting behaviour λ→ 0 as ||u||∞ → 1. This was initially observed in the numerical study of Pelesko (c.f. [38]). In Chapter 4 of this work I have concentrated on characterizing this final limiting branch of solutions. In the case of the beam problem and the annulus problem, this limiting behaviour is calculated explicitly by constructing the limiting solution branch as ||u||∞ → 1. In the case of the unit disc, an interesting observation is that the nature of the singularity at the origin has changed from u ∼ −1 + r2/3 in the membrane problem (1.1.2) to u ∼ −1 − 2r2 log r in the beam problem (1.1.4). It is conjectured that this change in singularity of the limiting solution is a necessary condition to destroy the infinite fold point structure. For the case of the fringing fields problem, a full characterization of the limiting behaviour as u→ −1 remains elusive. I believe this is a tractable problem but, despite its perhaps innocuous appearance, the δ|∇u|2 term induces a solution whose singularity structure is significantly more complex than that of the unperturbed problem. An explicit characterization of the break-up of the infinite fold points structure remains to be developed. An attempt to do so would require the analysis of equations 157 6.1. Micro-Electro Mechanical Systems (1.1.4)-(1.1.6) in the limit λ = O(1) with ε ≡ 1 − ||u||∞ → 0+ and δ → 0 where δ = δ0µ(ε) for some gauge µ(ε) satisfying µ → 0 as ε → 0. I believe these problems to be very challenging but ultimately resolvable by skillful use of matched asymptotic expansions. In § 4.4, the characterization of the maximal solution branch of the pure biharmonic MEMS problem (4.4.1) on a general geometry Ω ⊂ R2 is considered. Under the assump- tion that the solution concentrates at a unique x0 ∈ Ω as u(x0) → −1, the maximal branch is constructed and conditions on the point x0 are established in terms of the regular part of a Green’s function whose definition is given in (4.4.2). These results can be potentially expanded to consider certain domains for which the maximal solution branch may concentrate at multiple points in the domain, e.g. dumbbell shaped domains. It would be very interesting to rigorously establish the asymptotic results outlined here and in doing so definitively characterize the limiting behaviour of the biharmonic MEMS problems for general two dimensional geometries. The techniques implemented here have the potential to characterize concentration phenomena in other nonlinear eigenvalue problems which arise from the consideration of different physical situations. A few of these interesting problems are discussed in the following section. 6.1.1 Concentration in Other Nonlinear Eigenvalue Problems Bratu’s Problem is a description for the temperature u in a chemical reactor with geometry Ω ⊂ R3. For a spherical reactor of unit radius, the problem is formulated as d2u dr2 + 2 r du dr + λeu = 0, 0 < r < 1; u(1) = u′(0) = 0, (6.1.1) where the term eu arises from the Arrhenius rate law. The general properties of (6.1.1) are now well known. For example, it is well known that there exists a λc > 0 such that for λ > λc, (6.1.1) has no solutions ([25]). In addition, it is known that (6.1.1) exhibits an infinite fold point structure with limiting behaviour λ→ 2 as u(0)→∞ as developed in § 4.1.3 ( see also [12]). While some perturbations to this equation destroy this structure when introduced, other perturbations do not and leave the solution with essentially the same qualitative behaviour. For example, the problem d2u dr2 + 2 r du dr + λ exp [ u 1 + δu ] = 0, 0 < r < 1; u(1) = u′(0) = 0, (6.1.2) for δ > 0 has a bifurcation diagram with only a finite number of fold points and positive solutions for all λ > 0 (c.f. [2] ) indicating a significant deviation in character from the 158 6.1. Micro-Electro Mechanical Systems δ = 0 case. Alternatively, the problem d2u dr2 + 2 r du dr − δu + λeu = 0, 0 < r < 1; u(1) = u′(0) = 0, (6.1.3) retains all the qualitative features of (6.1.1) for δ > 0 including the infinite fold points structure and so does the problem d2u dr2 + 2 r du dr + λeu = 0, 0 < r < 1; u(1) = δu′(1), u′(0) = 0. (6.1.4) It is conjectured that the regularity of the limiting solution is key to determining the qualitative effects for each perturbation. It would be interesting to characterize more fully the quantitative and qualitative effects of the mentioned perturbed equations on the infinite fold point structure of the unperturbed equation (6.1.2). 6.1.2 Quenching Behavior in Fourth Order Time-Dependent MEMS A significant area for future work on MEMS devices relates to properties of the time dependent problem for u(x, t) where ∂u ∂t = −δ∆2u+∆u− λ (1 + u)2 , x ∈ Ω ; u = ∂nu = 0 x ∈ ∂Ω u(x, 0) = 0 (6.1.5) Here Ω is a bounded region of R2 with smooth boundary ∂Ω. Analysis from § 3.1.1 and [29] indicates that there is a critical value λ∗ such that if λ > λ∗, equation (6.1.5) does not have a steady state solution. In this case, it is expected that u will take the value −1 at some set of points in some finite time T . This phenomena is called touchdown. In the parlance of PDE theory, behaviour of this type is called quenching and corresponds to derivatives of the solution becoming unbounded. Recall that a solution blows up in a finite time T at a point x if |u(x, t)| → ∞ as t→ T− which is not the situation here as physical constraints demand that −1 ≤ u ≤ 0 for all time. The theory of singularity formation in second order differential equations is relatively well developed and accordingly touchdown in the case δ = 0 has been analyzed in some depth. In [70], the value of T was estimated and the resulting touchdown profile was characterized by means of a formal asymptotic expansion. These results have been rigorously verified in the work of [14] and [15] for several choices of f(x) and thus touchdown behaviour for the second order problem ut = ∆u− λf(x) (1 + u)2 , x ∈ Ω; u(x, t) = 0, x ∈ ∂Ω; u(x, 0) = 0, x ∈ Ω (6.1.6) 159 6.1. Micro-Electro Mechanical Systems is considered to be relatively well understood. Singularity formation in higher order problems is significantly more complicated and so presently the touchdown profile of the fourth order problem (6.1.5) has not been well characterized. One such complication is the observation that higher order problems may exhibit a countably infinite number of blowup profiles with only the observed solution being stable [3]. Numerical identification of such solutions is hampered by the presence of spurious positive eigenvalues generated as a byproduct of the similarity transforms [73], [74]. In the instance where Ω corresponds to the unit square, slab or unit disc, (6.1.5) may be discretized in space by replacing derivatives by differences quotients and in time by applying an explicit method ( e.g. Adams-Bashforth ) to the nonlinear term λ/(1 + u)2 and an implicit method to the differential term. The solution can be inte- grated relatively close to touchdown by making the time step sufficiently small, however, accurate determination of the touchdown profile very close to singularity requires special attention and has been considered in [73]. In the cases of the slab domain (−1, 1) and the unit disc {|x| ≤ 1}, it has been shown ([14, 15]) that for (6.1.6) touchdown is achieved only at the origin. The following numerical computations suggest that the inclusion of the bi-harmonic term , i.e. δ > 0 in (6.1.5), has a pronounced effect on touchdown behaviour, in particular touchdown may occur at multiple points for certain choices of Ω and regimes of λ, δ. 0 0.2 0.4 0.6 0.8 1 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 x u MEMS Deflection, λ = 2.0, δ = 0.003 0 0.2 0.4 0.6 0.8 1 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 x u MEMS Deflection, λ = 40.0, δ = 0.003 Figure 6.1: The left panel shows the dynamic deflection for λ = 2.0 and δ = 0.003 and touchdown at one point. Steady state is reached after approx- imately 0.3950 units of time. The right panel shows the dynamic deflection for λ = 40.0 and δ = 0.003. Touchdown occurs after 0.1190 units of time. In Fig. 6.1 we see that in the case of Ω = (0, 1) and δ = 0.003, equation (6.1.5) touches down at a single point for λ = 2 indicating that λ∗ < 2. In the case where λ = 40, touchdown occurs in finite time at two distinct points. This suggests the existence of some critical value λc(δ) satisfying λc(δ) > λ ∗(δ) and with the property that if λ > λc(δ), touchdown is achieved at two points while if λ ∗(δ) < λ < λc(δ), the 160 6.1. Micro-Electro Mechanical Systems beam touches down at a single point only. 0 0.2 0.4 0.6 0.8 10.51 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 x MEMS Deflection, λ = 5, δ = 0.003 y u 0 0.2 0.4 0.6 0.8 10.51 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 x MEMS Deflection, λ = 45, δ = 0.003 y u Figure 6.2: The left panel shows the configuration of the deflecting plate close to touchdown when δ = 0.003 and λ = 5. Touchdown occurs at the centre point (0.5, 0.5). The right panel shows the configuration of the deflecting plate close to touchdown when δ = 0.003 and λ = 45. In this case touchdown occurs at four distinct points. In Fig. 6.2, we see that in the case Ω = (0, 1)×(0, 1) and δ = 0.003, touchdown occurs at one point only when λ = 5 and at four distinct point when λ = 45 again suggesting the existence of some threshold λc(δ). The presence of four distinct touchdown points in the unit square domain suggests that this phenomena may be generated by the boundaries of the domain in some way. In the case of the unit disc ( not pictured ) touchdown is observed to occur on a small inner disc further suggesting that this phenomena is due to some boundary effect. To explain this multiple touchdown behaviour, it is instructive to consider the re- duced problem ut = −δuxxxx − λ (1 + u)2 , −1 < x < 1; u(±1, t) = ux(±1, t) = 0, u(x, 0) = 0 (6.1.7) in which (6.1.5) has been simplified by omitting the effects of tension and fixing Ω = (−1, 1). This problem is the simplest non-tailored ( f(x) = 1 ) MEMS deflection equa- tion which has been observed numerically to exhibit touchdown at multiple locations. Let us consider the situation at the centre of the beam for times close to t = 0. Assuming t is small enough, the only forces experienced by the centre of the beam should be that of the uniform electric field present in the gap between the plates. If that is the case, one might expect the term δuxxxx to be negligible in the centre section of the beam and thus it should deflect uniformly according to the equation ut = − λ (1 + u)2 , u(0) = 0 (6.1.8) 161 6.2. Eigenvalue Optimization Problems in Mathematical Ecology which has solution u(t) = −1+(1−3λt)1/3. In Fig. 6.3, the numerical solution of (6.1.7) is displayed along with the solution of (6.1.8) at four selected time steps. In the figure very good agreement is observed between these two quantities in the centre of the beam −1.0 −0.8 −0.6 −0.4 −0.2 0.0 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.0 0.25 0.5 0.75 1.0−0.25−0.5−0.75−1.0 x u u Figure 6.3: Numerical solutions of equation (6.1.7) for δ = 0.001 and λ = 100 at four time steps t = 0.000745, t = 0.002245, t = 0.002995 and t = 0.003244. The curves are arranged in order of increasing time from top to bottom. At each time step, the solution of (6.1.8) is overlaid. Numerical evidence lends weight to the suggestion that solutions of (6.1.7) for δ ≪ 1 are composed of a flat central region with governing equation (6.1.8) coupled to a propagating boundary effect. The case of multiple touchdowns then arises when the centre of the beam touches down before the boundary effect has time to propagate to the mid point. The dynamics of u after touchdown, i.e. t > T , may be studied in some weak sense where one might expect that u takes the value of −1 on open regions of Ω. This has been studied for blow-up phenomena in [11] in the context of fourth order reaction diffusion equations. In an actual MEMS device this would require some kind of insulating layer be present on the surface of the fixed plate to prevent the capacitor discharging from the contact between the two plates. 6.2 Eigenvalue Optimization Problems in Mathematical Ecology In Chapter 5, the two-term asymptotic expansion of the persistence threshold λ for the diffuse logistic model (1.2.1) in a highly patchy environment with spatially heteroge- neous growth rate (1.2.4) is calculated. The analysis permits for a relatively large class of spatially localized favourable and unfavourable habitats that are either interior to or on the boundary of a two-dimensional domain. The effect of habitat fragmentation on the coefficient of the leading-order term in the asymptotic expansion of λ is studied 162 6.2. Eigenvalue Optimization Problems in Mathematical Ecology and some general principles regarding the effect of fragmentation are established. These principles are summarized in Qualitative Results I–III of § 5.3.2. In certain degenerate cases, the optimization of λ requires the examination of the higher-order coefficient in the asymptotic expansion of λ. There are several interesting problems that warrant further study. First, it would be interesting to extend the single-species analysis to the case of multi-species interaction, such as predator-prey interactions, for which a partial fragmentation of the prey habitat may become beneficial for the persistence of the prey, rather than clumping the prey into a single habitat. This is based on the field observation that when one species is congregated, its predator has a distinct advantage and so the single species strategy may be sub-optimal. One possible avenue for investigating this multi-species case leads to the considera- tion of the diffusive Kermack-McKendrick Model ut = D1∆u+ u(m(x)− u)− βuv, vt = D2∆v − σv + µ+ βuv x ∈ Ω; ∂nu = ∂nv = 0, x ∈ ∂Ω (6.2.1) Here, D1,D2, σ, µ and β are positive constants. By investigating the linearized stability of the equilibrium (u, v) = (0, σ/µ), one is led to the following eigenvalue problem ∆u+ λu(m(x)− χ), x ∈ Ω; ∂nu = 0, x ∈ ∂Ω. (6.2.2) where χ = βσ/µ controls the strength of the predation. An asymptotic analysis of (6.2.1) may reveal that for certain m(x) a threshold χc can be established such that whenever χ > χc, fragmentation is optimal. Another interesting problem to be considered is that involving the so-called Allee effect (c.f. [32]), whereby individuals in a population find it difficult to locate a mate when their population is small. This effect is captured by the term a > 0 in ut = D∆u+m(x)u(1− u)(u+ a), x ∈ Ω; ∂nu = 0, x ∈ ∂Ω. (6.2.3) A schematic bifurcation diagram of equilibrium solutions to (6.2.3) is given in Fig. 6.4 where it is observed that the persistence threshold is now determined by a new critical value located by a fold point at the end of an unstable branch of solutions. 163 6.3. Conclusion λλ∗λ∗ ||u|| Unstable Stable Figure 6.4: The Allee effect. Schematic bifurcation diagram of (6.2.3) for a > 0 indicating the new persistence threshold λ∗. In this case, the optimal strategy is now determined by calculating and minimizing λ∗ rather than λ∗, as in the problem treated in Chapter 5. A third interesting avenue for future investigation involves the inclusion of more sophisticated mechanisms for population transport. For instance, suppose that individ- uals in a population not only diffuse, but advect along the gradient of the resources available to them. Under this assumption, the following advection diffusion model was proposed in [49], ut = ∇· [D∇u−αu∇m]+u(m−u), x ∈ Ω; ∂nu−αu∂nm = 0, x ∈ ∂Ω. (6.2.4) In this model, one might expect the geometry of the habitat to have a large effect on a fragmentation strategy as for non-convex domains, individuals will not be able to advect directly towards favourable resources when they are located in tight corners. Problems (6.2.1)-(6.2.4) are but a few of the many interesting ecological problems where the consequences of habitat fragmentation can be effectively analyzed with the techniques of matched asymptotic expansions. 6.3 Conclusion This thesis has been concerned with the implementation of contemporary singular per- turbation methods to two problems arising in the natural sciences and engineering. While asymptotic methods lack formal rigour, they have many other advantages which lend themselves well to the study of challenging mathematical problems. They can sys- tematically reduce difficult problems into a sequence of tractable ones which are readily solvable. For example, in Chapter 3 the location of the principal fold points of (1.1.4)- (1.1.6) were deduced based solely on knowledge of the fold point of the unperturbed problem (1.1.2). This technique avoids the need to solve additional nonlinear problems 164 6.3. Conclusion and so offers a significant simplification. In Chapter 4, asymptotic methods are used to resolve the λ/(1 + u)2 nonlinearity for ||u||∞ → 1 and provide an explicit solution characterization of the limiting solutions to MEMS problems. These results concern dynamically unstable solutions are therefore somewhat less important to practitioners, however, I hope they can provide the PDE community with avenues for further investigation. In Chapter 5, matched asymptotic expansions have been used to attack a long standing open problem in Mathematical Ecology. The work contained here does not resolve the question entirely as some particular assumptions have been made on the resource function m(x). 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