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Inverse and homogenization problems for maximal monotone operators Zarate, Ramon Saiz 2010

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Inverse and homogenization problems for maximal monotone operators by Ramón Zárate Sáiz 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) June 2010 c© Ramón Zárate Sáiz, 2010 Abstract We apply self-dual variational calculus to inverse problems, optimal control prob- lems and homogenization problems in partial differential equations. Self-dual variational calculus allows for the variational formulation of equa- tions which do not have to be of Euler-Lagrange type. Instead, a monotonicity condition permits the construction of a self-dual Lagrangian. This Lagrangian then permits the construction of a non-negative functional whose minimum value is zero, and its minimizer is a solution to the corresponding equation. In the case of inverse and optimal control problems, we use the variational functional given by the self-dual Lagrangian as a penalization functional, which naturally possesses the ideal qualities for such a role. This allows for the applica- tion of standard variational techniques in a convex setting, as opposed to working with more complex constrained optimization problems. This extends work pio- neered by Barbu and Kunisch. In the case of homogenization problems, we recover existing results by dal Maso, Piat, Murat and Tartar with the use of simpler machinery. In this context self-dual variational calculus permits one to study the asymptotic properties of the potential functional using classical Γ convergence techniques which are simpler to handle than the direct techniques required to study the asymptotic properties of the equation itself. The approach also allows for the seamless handling of multivalued equations. The study of such problems introduces naturally the study of the topological structures of the spaces of maximal monotone operators and their corresponding self-dual potentials. We use classical tools such as Γ convergence, Mosco con- vergence and Kuratowski-Painlevé convergence and show that these tools are well ii suited for the task. Results from convex analysis regarding these topologies are ex- tended to the more general case of maximal monotone operators in a natural way. Of particular interest is that the Γ convergence of self-dual Lagrangians is equiv- alent to the Mosco convergence, and this in turn implies the Kuratowski-Painlevé convergence of their corresponding maximal monotone operators; this partially ex- tends a classical result by Attouch relating the convergence of convex functions to the convergence of their corresponding subdifferentials. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Maximal monotone operators and variational methods for partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Inverse problems . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Homogenization theory . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Variational convergence . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Chapter contents . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Basics of convex analysis and the theory of monotone operators . . 8 2.1 Basic notation and setting . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Convex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 Fenchel-Young inequality . . . . . . . . . . . . . . . . . 10 2.2.2 ε-approximation of convex subdifferentials . . . . . . . . 10 2.2.3 Regularization of convex functions . . . . . . . . . . . . . 11 2.3 Maximal monotone operators . . . . . . . . . . . . . . . . . . . . 13 2.3.1 Regularization of maximal monotone operators . . . . . . 14 2.4 Convex potentials for maximal monotone operators . . . . . . . . 14 2.4.1 Integration of convex subdifferentials . . . . . . . . . . . 14 2.4.2 Fitzpatrick’s function . . . . . . . . . . . . . . . . . . . . 15 2.4.3 Regularization of convex functions on phase space . . . . 17 iv 3 Self-dual Lagrangians and their vector fields . . . . . . . . . . . . . 21 3.1 Self-dual Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Self-dual vector fields . . . . . . . . . . . . . . . . . . . . . . . . 22 3.3 Regularization of self-dual Lagrangians . . . . . . . . . . . . . . 22 3.4 Construction of self-dual potentials for a maximal monotone operator 23 3.5 ε-approximation of maximal monotone operators . . . . . . . . . 25 3.6 A variational principle . . . . . . . . . . . . . . . . . . . . . . . 27 3.7 Coercivity conditions . . . . . . . . . . . . . . . . . . . . . . . . 28 3.8 Space dependent maximal monotone operators . . . . . . . . . . 31 3.9 Self-dual Lagrangians onW 1,p0 (Ω)×W−1,q(Ω) . . . . . . . . . . 35 4 Topologies on spaces of convex Lagrangians and corresponding vec- tor fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.1 Mosco and Γ-convergence . . . . . . . . . . . . . . . . . . . . . 40 4.2 Kuratowski - Painlevé topology . . . . . . . . . . . . . . . . . . . 42 4.3 Mosco convergence vs Γ-convergence for self-dual functions . . . 43 4.4 Kuratowski-Painlevé convergence for maximal monotone operators 45 4.5 Continuity of the regularization of self-dual Lagrangians . . . . . 46 4.6 Continuity of regularizations of maximal monotone operators . . . 51 4.7 Attouch’s theorem: on the bicontinuity of the application f → ∂ f 51 4.7.1 Attouch’s theorem on phase space . . . . . . . . . . . . . 55 4.8 L 7→ ∂L and FT 7→ T are continuous . . . . . . . . . . . . . . . . 56 4.9 On the continuity of ∂L 7→ L . . . . . . . . . . . . . . . . . . . . 57 4.10 A compact class in Lp . . . . . . . . . . . . . . . . . . . . . . . . 58 5 A self-dual approach to inverse problems and optimal control . . . 63 5.1 An inverse problem via a variational, penalization based approach 63 5.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.3 Main results for optimal control problems . . . . . . . . . . . . . 68 5.4 Further generalizations and comments on coercivity and compact- ness conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6 Inverse and optimal control problems . . . . . . . . . . . . . . . . . 75 6.1 Identification of a convex potential and a transport term . . . . . . 75 v 6.2 Identification of a non conservative diffusion coefficient . . . . . . 80 6.3 Obtaining the cheapest temperature source control for a desired temperature profile . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.4 Identification of a convex potential and a transport term in a dy- namical setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.5 Identification of a non conservative diffusion coefficient in a dy- namical setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7 Homogenization of periodic maximal monotone operators via self- dual calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.1 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.2 Variational formula for the homogenized maximal monotone vec- tor field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.3 A variational approach to homogenization . . . . . . . . . . . . . 100 7.3.1 The homogenization of general Lagrangians onW 1,p(Ω)× Lq(Ω;RN) . . . . . . . . . . . . . . . . . . . . . . . . . . 100 7.3.2 Variational homogenization of maximal monotone opera- tors onW 1,p0 (Ω) . . . . . . . . . . . . . . . . . . . . . . . 113 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 vi Acknowledgments This work was carried out with the support of a fellowship from CONACYT (Con- sejo Nacional de Ciencia y Tecnologia) and the support offered by my advisor, Nassif Ghoussoub, to whom I am most thankful for his guidance, also. Special thanks must be given to Abbas Moameni, for the months spent working together where of great value. I am also grateful to Philip Loewen, of my supervisory committee, with whom I had fruitful discussions at various points. I thank the other members of my supervisory committee, Stephen Gustaphson and Richard Froese for their valuable efforts. Finally I want to specially thank Itzia, my wife, from whom I received immense peace and support during these last years. vii Chapter 1 Introduction This work deals with the application of variational techniques to inverse and ho- mogenization problems for certain types of partial differential equations. Even if the approach is variational, these equations do not have to be of Euler-Lagrange type (the classical variational setting), rather the variational principle used here re- lies on Fenchel-Young duality. Examples include, but are by no means limited to, skew adjoint vector fields superimposed on a gradient of some vector potential, and parabolic equations. This variational theory and its applicability to solve partial differential equa- tions was developed in a series of papers by Ghoussoub, culminating in the book [Gho08a], and will be the main tool. The techniques involve a dose of convex analysis on phase space, which is not a typical technique in the study of partial differential equations, but the approach unifies under the same theory the study of a big class of partial differential equations. One of the goals of this work is to show how this unified theory can be used and applied to the study of inverse and homogenization problems associated to such families of equations. The connection between inverse and optimal control problems and homoge- nization problems is that all of them call for the study of asymptotic properties of the families of equations, in particular of their topological structure. The varia- tional setting provides interesting, well understood tools for such a purpose, and which are compatible with the spirit of generality of the theory. 1 Let us identify some individual topics involved in this work. 1.1 Maximal monotone operators and variational methods for partial differential equations Monotone operators and maximal monotone operators were introduced as non lin- ear generalizations of linear positive definite operators. They were studied in detail by many authors, including Brezis, Browder, Nirenberg and Rockafellar, among others. They now constitute an important tool for modern non linear analysis. Their study is classically connected with convex analysis, since they can be considered as extensions of subdifferentials of convex functions: it is a well known fact that convex subdifferentials are themselves maximal monotone operators. Of more recent origin is the fact that the connection goes beyond that: The monotonicity condition can actually be seen as a convexity condition on phase space, and this can be used to construct convex potentials on this larger space that “realize” a specific vector field through a variational principle, without being constrained to being the Euler-Lagrange equation of any potential. This variational principle has two ingredients: First, and most obvious, is the minimization of certain functional. Second, the requirement that the minimum of such a functional has to be a specific value (which is naturally normalized to zero). It is in this context that the condition of self-duality is chosen in the work of Ghoussoub, as it actually grants the needed conditions for this value to be achieved, without a priori knowledge of existence of solutions. This very simple feature (the fact that the value of the minimized functional is actually zero) will be the main idea to exploit in the development of a general variational approach to inverse and optimal control problems. 1.2 Inverse problems Partial differential equations are a prime tool for modelling nature. A lot of en- gineering, physical and biological problems are modelled with partial differential equations. A direct problem for partial differential equations consists of, given specified parameters which represent properties in the modelled phenomena, finding a so- 2 lution to the equation corresponding to those parameters, and thus, predicting the behaviour of such a system from the knowledge of its parameters. An inverse problem consists of, given a solution for the partial differential equation, that is, a known behaviour of the modelled system, to recover the pa- rameters defining the system. Inverse problems allow, in principle, to recover hid- den or internal data from a system given experimental observations from it; this plays a fundamental role in modern science and engineering. They can be used in medicine, where, for example, Magnetic Resonance Imaging is used to recover information on the inner tissue of a patient without direct observation that would require a surgical procedure; inverse spectral methods are used in astrophysics to study the composition of stellar bodies which are completely impossible to explore by direct means. Inverse problems can also be used in the design of devices where the interaction of inner mechanics are understood, but one needs to control the be- haviour of such devices by the appropriate selection of parameters that represent design decisions. Another typical example is the exploration of oil and mineral deposits under the crust of the earth, minimizing needless and costly excavation. Needless to say, inverse problems are mathematically very challenging by their very nature. The starting point of the theory developed here is the work by Barbu and Ku- nisch, where they approach the problem via a least squares approach. The difficulty to such approaches is that the study of solution manifolds to families of partial dif- ferential equations is inherently complicated. To ease this difficulty the approach consists of using an appropriate functional that penalizes parameters not being in such solution set, but allowing us to work in the original ambient spaces which are typically convex, and thus allow for the use of classic variational techniques. Then as the cost of this penalization grows, one expects to find gradually better ap- proximations to solutions to the system that minimize the penalized least squares problem. This is where the variational principle mentioned above plays a fundamental role, as this functional is constructed in such a way that it is itself an excellent penalization to the solution set, being a positive functional whose zeroes are the solutions to the equations. 3 1.3 Homogenization theory Homogenization theory for partial differential equations is the study of asymp- totic behaviour of families of partial differential equations. A typical setting is when complicated behaviour in the system (say, highly oscillating parameters) can be approximated by simple behaviour (averaging of such parameters, giving ho- mogenized simpler parameters), but one then needs to formally justify that such approximation is consistent. It is not enough that equations are approximated, one needs to show that the corresponding solutions are also being approximated. Classically, for the case of variational equations (equations of Euler-Lagrange type) the potential functionals are easier to study, and from their limiting behaviour one deduces the behaviour of their minimizers, thus obtaining information about the asymptotic behaviour of the equations and solutions themselves. This obvi- ously fails as soon as the equations involved are not variational. The approach then required is more technical and complicated. In particular the homogenization of monotone vector fields has been widely studied by non variational techniques by many authors, including dal Maso, Murat, Tartar and others. Here we use the more general variational setting and show that traditional variational proofs, simpler in nature, work just as well for these cases. The principal idea here is that the topological structure of monotone families is tightly connected with the topological structure of their corresponding potentials. The study of such topological properties is an important part of this work. 1.4 Variational convergence Variational convergence refers to the study of convergence properties of function- als where the notion of convergence translates well to the convergence of their minimizers. A very classical notion of convergence for spaces of compact convex sets is the Hausdorff metric. In order to study the topological structure of a space of convex functionals, it is of advantage to identify these functionals with their epigraphs, which are convex sets. The compactness criteria, however, is an unac- ceptable limitation in this case. A more suitable choice of convergence criteria of general sets is that of Kuratowski-Painlevé convergence. This notion, adapted to infinite dimensional spaces and their inherent topologies, gives rise to Γ (Gamma) 4 and Mosco convergence, both already classical tools in functional analysis, known to be appropriate notions of convergence to functionals, and particularly well be- haved when working with convex families. To study the convergence properties of families of monotone maps, the raw notion of Kuratowski-Painlevé convergence can be used and has been shown to be the right choice by the work of authors like Kato, Brezis, Damlamian, Attouch and others. Of particular interest is the fact that for the case of convex potentials and convex subdifferentials, Mosco convergence of the potentials corresponds to Kuratowski-Painlevé convergence of their subdif- ferentials (modulo a normalizing condition). In the more general context of maxi- mal monotone operators a lot of the correlations among the topological structure is preserved, in particular, the convergence of a family of the convex functionals as- sociated to a family of maximal monotone operators, does imply the convergence of such operators. This is the heart of the homogenization result presented here. 1.5 Chapter contents Parts of this work are taken from [GMZ10], which is joint work with Nassif Ghous- soub and Abbas Moameni. The layout of the work is as follows: In Chapter 2 the basic notions of convex analysis and the theory of maxi- mal monotone operators that will be used throughout this work are introduced. How maximal monotone operators can be represented by a certain convex poten- tial (Fitzpatrick’s function) and the general notion of associating convex potentials to monotone operators is introduced here, which is one of the principal ingredients for the later parts of the work. Also a standard regularization technique for convex functions and for maximal monotone operators is introduced. In Chapter 3 a special class of convex potentials and its properties are intro- duced: the class of self-dual Lagrangians. We show how these special potentials can be constructed using the framework of Chapter 2. Also, a few important gener- alizations of classical convex analysis results to general maximal monotone oper- ators are introduced here, for later use in the work, such as a generalization of the celebrated Brønsted-Rockafellar Lemma. The main variational notion to be used later is introduced here, where a positive functional is constructed in such a way that its zeroes correspond to solutions to a given partial differential equation. This 5 will play a central role in the treatment of inverse and optimal control problems. Also we describe how to deal with space dependent operators appearing in par- tial differential equations and show their respective self-dual Lagrangians, which allow certain types of non variational equations to be solved using the aforemen- tioned variational principle; this will be the main setting for the homogenization result presented here. Chapter 4 is devoted to the study of specific topologies for spaces of con- vex functionals and spaces of maximal monotone operators and their connections through the theory introduced earlier. The notions involved are the already men- tioned Γ, Mosco and Kuratowski-Painlevé convergence. We show how Mosco and Γ convergences are the same mode of convergence when restricted to the space of self-dual Lagrangians, and show how this mode of convergence of potential functionals implies Kuratowski-Painlevé convergence of their associated maximal monotone operators; this result is a partial generalization of Attouch’s Theorem, which states the equivalence of the notions when restricted to convex subdifferen- tials modulo a normalizing condition. We include here a slight reformulation of the Theorem on phase space which removes the need for any normalizing condition. The converse implication, which does not hold, at least without some restrictions, is briefly discussed. Additionally, we include in this chapter various classical and some new results regarding the continuity of the regularization notions for convex functions and monotone operators alike. The final section of this chapter contains simple compactness results in a more specific setting for special classes of func- tionals for later reference in chapter 6. Chapter 5 contains the main results regarding inverse and optimal control problems. The result is stated in an abstract setting to emphasize on its generality. It provides an unified approach under very general topological conditions for ex- istence of optimal least squares solutions, and then generalized to optimal control problems. Several versions of the results are presented to briefly outline general situations regarding the required compactness, but the results are also intended as general guides and can be adapted to other situations not considered here for the sake of simplicity. The approach consists of minimizing a positive functional con- structed via the variational functional from the variational principle of chapter 3 used as a penalization. This constructed functional is defined in the whole ambient 6 space associated to the given equation, which removes the need to work in the man- ifold of solutions generated by the parameters; these minimizers are then shown to be approximate solutions which then converge to the optimal true solution to the least squares problem by making the penalty parameter increase. Then, in Chapter 6, sample partial differential equations applications of vari- ous types are provided to illustrate the applicability of the results from the previous chapters, as well as to illustrate its broad scope in terms of the different possible approaches for parametrizing the families of equations. This chapter also outlines how certain partial differential equations are put in the context of self-dual La- grangians. Finally Chapter 7 contains a homogenization result where the variational set- ting is used to study the asymptotic behaviour of a specific family of equations parametrized by the period of a given, possibly multivalued, monotone, space- dependent non-linearity. These non-linearities can be assumed to be non-variational, in the sense that they do not correspond to the potential of a functional. The result recovered is a known one, but the novelty lies in the use of more direct homog- enization results for convex functionals similar to those needed for the classical variational setting. The periodic setting is chosen for simplicity and to be able to easily express the limit functionals and equations, but the approach is expected to be usable in more general settings. 7 Chapter 2 Basics of convex analysis and the theory of monotone operators In this chapter we present some of the basics of convex analysis and the theory for maximal monotone operators. It is provided for convenience of the reader, and as reference, in the interest of making this work as self contained as possible. Convex analysis and maximal monotone operators are traditionally related by the fact that maximal monotone operators are known to be generalizations of con- vex subdifferentials. Their connection however goes deeper than that, and some aspects will be revealed in this and following chapters. Reference on the topics for this chapter can be found in [Roc70, Att84, BP86, Phe93, ET99]. 2.1 Basic notation and setting From here on, X will denote a reflexive Banach space and X∗ will denote its dual. The space X×X∗ will be referred to as phase space. When working in this abstract setting, we will use x,y,z to denote variables in X , and p,q,r to denote variables in X∗. ‖ · ‖ will be used to denote the norms in X and X∗. The duality pairing between x ∈ X and p ∈ X∗ (that is, the value of the func- tional p at x) will be denoted as 〈x, p〉. 8 The duality mapping is the (possibly multivalued) map J : X ⇒ X∗ defined by Jx := {p ∈ X∗ : ‖p‖= ‖x‖ and 〈x, p〉= ‖x‖2}; we will assume that the Banach space X is uniformly convex, in such a way that the mapping J is single valued. This mapping is known to be invertible whenever X is reflexive (see [BP86, Proposition 2.16]). It is also important to point out that this mapping is not linear, in general (only in the Hilbert case). 2.2 Convex analysis A function f : X → R∪{+∞} is said to be convex if for any 0 ≤ λ ≤ 1, and any x,y ∈ X , f (λx+(1−λ )y) ≤ λ f (x)+ (1−λ ) f (y). It is said to be proper if f is not identically +∞. The function f is said to be lower semi continuous if the epigraph of f : epi( f ) := {(x,λ ) ∈ X×R : λ ≥ f (x)} is a closed set. The class of convex, proper, lower semi continuous functions on a Banach space X will be denoted by Γ(X). Consider f : X →R∪{+∞}, a proper lower semi continuous convex function. The subdifferential mapping ∂ f : X ⇒ X∗ is the multivalued mapping defined by ∂ f (x) := {p ∈ X∗ : f (y)≥ f (x)+ 〈y− x, p〉, ∀y ∈ X}. The duality mapping J can be seen as a subdifferential: Jx= ∂ (‖·‖2 2 ) (x), and simi- larly, its inverse J−1 : X∗→ X corresponds to J−1p= ∂(‖·‖22 )(p). The Fenchel-Legendre transform (or conjugate) of f , denoted by f ∗, is the 9 convex function f ∗ : X∗→ R∪{+∞} defined by f ∗(p) := sup x∈X {〈x, p〉− f (x)}. The function f ∗ is a proper lower semi continuous convex function on X∗. f is proper convex lower semi continuous if and only if f ∗∗ = f . 2.2.1 Fenchel-Young inequality The following important inequality, referred to as the Fenchel-Young inequality, holds: f (x)+ f ∗(p)≥ 〈x, p〉. (2.1) The case when equality is achieved in the above can be characterized the fol- lowing way: f (x)+ f ∗(p) = 〈x, p〉 ⇐⇒ p ∈ ∂ f (x) ⇐⇒ x ∈ ∂ f ∗(p). (2.2) The above will be referred to as the duality case in the Fenchel-Young inequality. Using Fenchel-Young duality, one can parametrize the subdifferential mapping the following way: ∂ f (x) := {p ∈ X∗ : f (x)+ f ∗(p)−〈x, p〉= 0}. (2.3) We will denote the graph of the subdifferential of a convex function f by ∂ f , that is: ∂ f := {(x, p) ∈ X×X∗ : p ∈ ∂ f (x)}. 2.2.2 ε-approximation of convex subdifferentials From (2.3), and considering inequality (2.1), it seems natural to think of a point (x, p) to be approximating ∂ f if f (x)+ f ∗(p)−〈x, p〉 ≤ ε . This notion is formal- ized in the following definition. Definition 2.1 The ε-subdifferential of a convex function at a point x ∈ X, denoted 10 ∂ε f (x), is defined by ∂ε f (x) := {p ∈ X∗ : f (x)+ f ∗(p)−〈x, p〉 ≤ ε}. It will be also denoted: ∂ε f := {(x, p) ∈ X×X∗ : p ∈ ∂ε f (x)}. The ε-subdifferential is a useful notion of a “perturbation” or approximation to a convex subdifferential. How near a point in ∂ε f lies to a point in ∂ f is made precise by the following celebrated result, best known as the Brøndsted-Rockafellar Lemma: Theorem 2.2 If p0 is an ε-subgradient of f at x0, (i.e. p0 ∈ ∂ε f (x0)), then for any λ > 0 there exists a pair (xλε , pλε ) ∈ ∂ f such that • ‖xλε − x0‖ ≤ √ ε λ • ‖pλε − p0‖ ≤ λ √ ε This result appears in [BR65], a proof based on Ekeland’s variational principle can be found in [Bee93]. The proof of the above is also contained in the proof provided here for Theorem 3.9. 2.2.3 Regularization of convex functions Here is an important way to combine convex functions. Definition 2.3 Given f ,g ∈ Γ(X), their inf-convolution, denoted by f ? g, is the mapping given by f ?g(x) := inf y∈X { f (y)+g(x− y)}. The interaction of the inf-convolution with the Fenchel-Legendre conjugate is described by the next well known result (see for example [Att84]). Proposition 2.4 Let f and g be functions in Γ(X). Then ( f ?g)∗ = f ∗+g∗. 11 If furthermore, dom( f )−dom(g) contains a neighbourhood of the origin, then also ( f +g)∗ = f ∗ ?g∗. Inf-convolution with the square of the norm produces the Moreau-Yosida reg- ularization of a convex function. Definition 2.5 Let f be a function in Γ(X). For each λ > 0, the function fλ (x) := inf y∈X { f (y)+ ‖x− y‖ 2 2λ } is the Moreau-Yosida regularization of f with parameter λ . The functions fλ approximate f in the sense that (for f ∈ Γ(X)): f (x) = lim λ→0 fλ (x) = sup λ>0 fλ (x) ∀x ∈ X , also fλ is locally Lipschitz (see [Att84]). The regularization also possesses addi- tional boundedness conditions: it is easy to see that fλ (x)≤ f (0)+ ‖x‖ 2 2λ . The following is a basic property (see [Att84, Proposition 2.68]): Proposition 2.6 Let f ∈ Γ(X) and let λ1, λ2 be positive constants. We have( fλ1 ) λ2 = fλ1+λ2 . The above can be shown from the following fact, which will be used later in the proof of a similar proposition. Lemma 2.7 For λ1, λ2 > 0: inf z∈X {‖x− z‖ 2λ1 2 + ‖y− z‖ 2λ2 2 }= 1 2(λ1+λ2) ‖x− y‖2. Proof: The minimizer above is reached at a point z∗ ∈ X such that 1λ1 (z∗− x) = 1 λ2 (y− z∗), from this we compute z∗ = λ1λ2 λ1+λ2 ( x λ1 + y λ2 ) . Then z∗− y= λ1λ1+λ2 (x− y) 12 and x− z∗ = λ2λ1+λ2 (x− y). Using the above simply compute ‖x− z∗‖ 2λ1 2 + ‖y− z∗‖ 2λ2 2 = 1 2(λ1+λ2) ‖x− y‖2.  2.3 Maximal monotone operators It is convenient to identify a multivalued operator T : X ⇒ X∗ with its graph in X×X∗, this means that T will denote the set {(x, p) ∈ X×X∗ : p ∈ Tx}. Definition 2.8 A multivalued operator T : X ⇒ X∗ is said to be monotone if for any (x, p) ∈ T and any (y,q) ∈ T the following holds: 〈x− y, p−q〉 ≥ 0. If furthermore, T cannot be extended by another monotone operator, then T is said to be maximal monotone. It is a known result due to Rockafellar that a convex subdifferential is itself a maximal monotone operator; not all maximal monotone operators, however, are convex subdifferentials. We will come back to this point later on. The following result, known as Minty’s Theorem in the Hilbert setting and ex- tended to reflexive Banach spaces by Rockafellar, is an important characterization of maximal monotone operators: Theorem 2.9 Let T : X ⇒ X∗ be a monotone operator. The following are equiva- lent: • T is maximal monotone. • For each λ > 0, the operator T + 1λ J is surjective. 13 2.3.1 Regularization of maximal monotone operators According to the previous characterization, after translation, to each x ∈ X there exists a unique solution, z, to J(z− x)+λTz 3 0. (2.4) Definition 2.10 The resolvent of T with parameter λ is the operator RTλ : X → X given by RTλ x := z where z is given by (2.4). The Yosida approximation of T with parameter λ is the mapping Tλ : X → X∗ given by Tλ x := 1 λ J(x−RTλ x). The above operators RTλ and Tλ are known to be continuous (see [Att84]). Con- vergence properties will be stated in chapter 4. Remark 2.11 Observe that Tλ x ∈ T ( RTλ x ) . In the case of convex subdifferentials, the following relation between the regu- larized operator and the regularized convex potential holds. Proposition 2.12 Let f ∈ Γ(X), then R∂ fλ x= argmin y∈X { f (y)+ ‖x− y‖ 2 2λ }. That is: fλ (x) = f (R ∂ f λ x)+ ‖(R∂ fλ x)−x‖2 2λ . Also ∂ ( fλ ) = ( ∂ f ) λ . 2.4 Convex potentials for maximal monotone operators 2.4.1 Integration of convex subdifferentials As mentioned above, convex subdifferentials are maximal monotone operators, however, not every maximal monotone operator is a convex subdifferential. This can be made precise by using the following definition. 14 Definition 2.13 A multivalued operator T : X ⇒ X∗ is said to be cyclically mono- tone if 〈x0− x1, p0〉+ · · ·+ 〈xn−1− xn, pn−1〉+ 〈xn− x0, pn〉 ≥ 0 for any chain (xi, pi) ∈ T , i = 0, . . . ,n. It is said to be maximal if in addition its a maximal monotone operator. The following is an important result by Rockafellar, it classifies convex subd- ifferentials among the class of maximal monotone operators and provides an inte- gration formula to recover a convex function from its subdifferential. Theorem 2.14 A maximal monotone mapping T is cyclically maximal monotone if and only if there exist a convex function f such that T = ∂ f . Furthermore, by fixing x0 ∈ dom(T ), f can be recovered by the formula f (x) = f (x0)+ sup{〈x− xn, pn〉+ · · ·+ 〈x1− x0, p0〉}, where the sup above is taken over all finite sets {(xi, pi)}ni=1 with (xi, pi) ∈ T , i = 0, . . . ,n. Remark 2.15 Observe that the value f (x0) could be chosen arbitrarily. Convex functions are characterized by their subdifferentials only up to an additive con- stant. In other words: ∂ f = ∂g if and only if for some constant c, f = g+ c. 2.4.2 Fitzpatrick’s function Let us make a simple observation in the monotonicity condition, T is monotone if for (x, p),(y,q) ∈ T : 0≤ 〈x− y, p−q〉= 〈x, p〉+ 〈y,q〉−〈x,q〉−〈y, p〉, this can be rewritten as 〈(x, p),(y,q)〉X×X∗−〈y,q〉 ≤ 〈x, p〉. The left hand side above, fixing (y,q)∈ T , can be seen as a linear functional on X× X∗. This is effectively a convexity condition in which the following construction is 15 based: Definition 2.16 Given T a maximal monotone operator, its Fitzpatrick function, FT : X×X∗→ R∪{∞} is defined by FT (x, p) := sup (y,q)∈T {〈x,q〉+ 〈y, p〉−〈y,q〉}. This construction plays a fundamental role in the relation between the theory of maximal monotone operators and convex analysis. It appeared originally in [Fit88]. The importance of Fitzpatrick’s function is that it provides a counterpart “integration formula” for maximal monotone operators. The cyclical monotonicity condition on a vector field can be made equivalent to that of conservative. Other maximal monotone vector fields are not integrable, in this sense. The properties of FT are summarized in the following proposition (see [Gho08a]). Proposition 2.17 If T is maximal monotone, then FT is a proper convex function on X×X∗ and the following inequalities hold: 〈x, p〉 ≤ FT (x, p)≤ F∗T (p,x). Furthermore: 〈x, p〉= FT (x, p) = F∗T (p,x) ⇐⇒ (x, p) ∈ T (2.5) Remark 2.18 Compare (2.5) with (2.2). The maximal monotone operator T can be parametrized as T = {(x, p) ∈ X×X∗ : FT (x, p) = 〈x, p〉} (2.6) which is an analogue to (2.3). The above justifies the notion that the function FT is a “convex representative” of the operator T . 16 Definition 2.19 A function F is said to be a representative for the operator T if both: 1. F(x, p)≥ 〈x, p〉. 2. F(x, p) = 〈x, p〉 if and only if (x, p) ∈ T . The following lemma appears in [SZ04, Lemma 1.1]. Lemma 2.20 Let T be a maximal monotone operator. If (x, p) and (y,q) are ele- ments of X×X∗ such that (x, p) ∈ ∂FT (y,q), then 〈x− y, p−q〉 ≤ inf (z,r)∈T 〈x− z, p− r〉. In particular 〈x− y, p−q〉 ≤ 0, and 〈x− y, p−q〉= 0 if and only if (x, p) ∈ T . 2.4.3 Regularization of convex functions on phase space A direct analogue of the Moreau-Yosida regularization on phase space would sim- ply be, for F ∈ Γ(X×X∗): Fλ (x, p) := inf (y,q)∈X×X∗ {F(y,q)+ ‖x− y‖ 2+‖p−q‖2 2λ }. The following are other ways of regularizing a convex function on phase space: Definition 2.21 Let F be a function in Γ(X ×X∗). Take λ > 0. Define the follow- ing: • Fλ (x, p) := inf(y,q)∈X×X∗{F(y,q)+ ‖x−y‖ 2 2λ + λ 2 ‖p−q‖2}. • Fλ1 (x, p) := infy∈X{F(y, p)+ ‖x−y‖ 2 2λ + λ 2 ‖p‖2}. • Fλ2 (x, p) := infq∈X∗{F(x,q)+ ‖x‖ 2 2λ + λ 2 ‖p−q‖2}. • Fλ1,2(x, p) := inf(y,q)∈X×X∗{F(y,q)+ ‖x−y‖ 2 2λ + λ 2 ‖p‖2+ λ2 ‖p−q‖2+ ‖y‖ 2 2λ }. Remark 2.22 Fλ1,2(x, p) = ( Fλ2 )λ 1 (x, p). 17 The analogue to proposition 2.6 is the following Proposition 2.23 For λ1, λ2 > 0, denote λ1 ?λ2 := λ1λ2λ1+λ2 . We have( Fλ11 )λ2 1 = F λ1+λ2 1 and ( Fλ12 )λ2 2 = F λ1?λ2 2 To prove this we will use lemma 2.7 and its following analogue: Lemma 2.24 For λ1, λ2 > 0: inf r∈X∗ {λ1 2 ‖p− r‖2+ λ2 2 ‖q− r‖2}= λ1 ?λ2 2 ‖p−q‖2. Proof: The minimizer above is a point r∗ ∈ X∗ such that λ1(r∗− p) = λ2(q− r∗), from this we compute r∗ = λ1p+λ2qλ1+λ2 . Then r∗−q= λ2λ1+λ2 (p−q) and p− r∗ = λ1λ1+λ2 (p−q). Using the above simply compute λ1 2 ‖p− r∗‖2+ λ22 ‖q− r∗‖ 2 = λ1 ?λ2 2 ‖p−q‖2.  Proof of proposition 2.23: Simply write ( Fλ11 )λ2 1 (x, p) = infz∈X {Fλ11 (z, p)+ ‖x− z‖2 2λ2 + λ2 2 ‖p‖2} = inf z∈X inf y∈X {F(y, p)+ ‖z− y‖ 2 2λ1 + λ1 2 ‖p‖2+ ‖x− z‖ 2 2λ2 + λ2 2 ‖p‖2} = inf y∈X { F(y, p)+ inf z∈X {‖z− y‖ 2 2λ1 + ‖x− z‖2 2λ2 }+ λ1 2 ‖p‖2+ λ2 2 ‖p‖2}, 18 which using lemma 2.7 gives ( Fλ11 )λ2 1 (x, p) = infy∈X {F(y, p)+ 1 2(λ1+λ2) ‖x− y‖2+ λ1+λ2 2 ‖p‖2} = Fλ1+λ21 (x, p). The second statement follows in an analogous fashion, using lemma 2.4.3.  The following lemma in the spirit of proposition 2.12 regards the first regular- ization type and Fitzpatrick’s function. Lemma 2.25 Let (x, p) ∈ X×X∗ and λ > 0 be such that ( FT )λ (0,0) = FT (x, p)+ 12λ ‖x‖2+ λ2 ‖p‖2. Then (x, p) ∈ T , and p=− 1λ Jx. That is: x= RTλ0 and p= Tλ0. Proof: If (x, p) is as above, then we have that (0,0) ∈ ∂FT (x, p)+∂ ( 1 2λ ‖x‖2 ) +∂ (λ 2 ‖p‖2 ) . Then there must be some (y,q) such that (q,y) ∈ ∂FT (x, p) and (−q,−y) = ( 1 λ Jx,λJ−1p). (2.7) From lemma 2.20, we have that 0 ≥ 〈x− y, p−q〉 = 〈x,−q〉+ 〈−y, p〉+ 〈x, p〉+ 〈y,q〉 = 1 λ ‖x‖2+λ‖p‖2+ 〈x, p〉+ 〈y,q〉. Also, since 〈x, p〉 ≥ −‖x‖‖p‖ and 〈y,q〉 ≥ −‖y‖‖q‖=−‖x‖‖p‖, we get 〈x− y, p−q〉 ≥ (√λ‖x‖− 1√ λ ‖p‖)2 ≥ 0. 19 Hence all inequalities are equalities above, we obtain that (y,q) ∈ T , but also 1√ λ ‖x‖= √ λ‖p‖= 1√ λ ‖y‖= √ λ‖q‖, and 〈y,q〉=−‖ 1√ λ y‖‖ √ λq‖; these equalities imply that − 1√ λ y= J−1 √ λq, that is − 1λ Jy= q, which implies, in view of (2.7), that (y,q) = (x, p). We can conclude (x, p) ∈ T and p=− 1λ Jx.  20 Chapter 3 Self-dual Lagrangians and their vector fields The notion of a self-dual Lagrangian is introduced here, even if the self-duality condition is not required in order to represent a given monotone operator (see defi- nition 2.19) in many practical examples self-duality can be achieved naturally. It is also a condition that will be sufficient to guarantee existence of solutions for equa- tions expressed by monotone operators via the variational principle introduced by Ghoussoub (without a priori knowledge of the existence of such solutions by dif- ferent means). 3.1 Self-dual Lagrangians Definition 3.1 A function L ∈ Γ(X×X∗) is said to be a self-dual Lagrangian if for any (x, p) ∈ X×X∗ it satisfies L(x, p) = L∗(p,x), where L∗ denotes the Fenchel-Legendre transform in X×X∗. The class of self-dual Lagrangians on X×X∗ will be denoted byL (X). 21 The Fenchel-Young inequality gives that L(x, p)+L∗(p,x)≥ 〈(x, p),(p,x)〉= 2〈x, p〉. From self-duality this is: L(x, p)≥ 〈x, p〉. Every self-dual Lagrangian then obeys: L(x, p)≥ 〈x, p〉, ∀(x, p) ∈ X×X∗. (3.1) Note that self-dual Lagrangians can be translated: considerM(x,q) := L(x, p+ q)− 〈x, p〉 or N(y, p) := L(x+ y, p)− 〈x, p〉, which are themselves self-dual La- grangians whenever L is. 3.2 Self-dual vector fields The previous relation (3.1), in consideration of (2.3) and (2.6), motivates the fol- lowing definition. Definition 3.2 Given a self-dual Lagrangian L, the associated self-dual vector field is the operator ∂L : X ⇒ X∗ given by ∂Lx := {p ∈ X∗ : L(x, p) = 〈x, p〉}. The following is of fundamental importance (see [Gho08a]): Proposition 3.3 For any L ∈L (X), the operator ∂L is maximal monotone. It can be seen now, that self-dual Lagrangians are potentials to (at least some) maximal monotone operators. The converse holds: it is possible to find a self- dual Lagrangian representing a given maximal monotone operator. This will be addressed in section 3.4. 3.3 Regularization of self-dual Lagrangians Refering to definition 2.21, the following proposition is noteworthy. It states that self-dual Lagrangians can be regularized while preserving self-duality. Proposition 3.4 If L : X ×X∗ → R∪ {+∞} is a self-dual Lagrangian, then for each λ > 0, also Lλ1 ,L λ 2 and L λ 1,2 are self-dual Lagrangians. The above appears in [Gho08a, Lemma 3.2]. 22 3.4 Construction of self-dual potentials for a maximal monotone operator Fitzpatrick’s function allows for a simple construction of a self-dual convex poten- tial for any maximal monotone operator, by means of the proximal average of the Fitzpatrick’s function and its conjugate. Definition 3.5 Consider a maximal monotone operator T : X ⇒ X∗, fix a> 1 and let b be its conjugate exponent, that is: 1a + 1 b = 1. We define the index a self-dual potential corresponding to T the following way: LaT (x, p) = inf {1 2FT (x1, p1)+ 1 2F ∗ T (p2,x2)+ 1 4a‖x1− x2‖a+ 14b‖p1− p2‖b : (x, p) = 12(x1, p1)+ 1 2(x2, p2) } . (3.2) A standard choice is a= 2= b, and thus we will denote: LT := L2T . The properties of LaT are summarized in the following Proposition 3.6 Let T : X ⇒ X∗ be maximal monotone. Then for each a > 1, LaT is a proper lower semi continuous convex function on X×X∗ and LaT (x, p) = ( LaT )∗(p,x) (i.e. LaT ∈L (X)), LaT (x, p)≥ 〈x, p〉. Furthermore: 〈x, p〉= LaT (x, p) ⇐⇒ (x, p) ∈ T (3.3) For the proof we will use the following fact (see [Gho08a, part (11.) of Proposition 2.6]): For h(x) := inf{F(x1,x2) : x = x1+x22 }, where F ∈ Γ(X ×X), its Fenchel- Legendre conjugate is given by h∗(p) = F∗( p2 , p 2 ). Proof of proposition 3.3: We will show the self-duality of LaT first, using the previous fact. Denoting G1 ( (x1, p1),(x2, p2) ) := 12FT (x1, p1) + 1 2F ∗ T (p2,x2), and 23 G2 ( (x1, p1),(x2, p2) ) := 14a‖x1−x2‖a+ 14b‖p1− p2‖b.Hence, using the fact above:( LaT )∗(p,x) = (G1+G2)∗(( p2 , x2),( p2 , x2)). Direct computation yields G∗1 ( (p1,x1),(p2,x2) ) = 12F ∗ T (2p1,2x1)+ 1 2FT (2x2,2p2), and G∗2 ( (p1,x1),(p2,x2) ) = { 4a−1 a ‖x1‖a+ 4 b−1 b ‖p1‖b (x1, p1) = (−x2,−p2) +∞ else. Since at least G2 is continuous, the conjugate of the sum is the inf-convolution of the conjugates, giving that ( LaT )∗(p,x) = (G∗1 ?G∗2)(( p2 , x2),( p2 , x2)) = inf (y1,q1),(y2,q2) {1 2 F∗T (2q1,2y1)+ 1 2 FT (2y2,2q2)+ + G∗2 ( (q1− p2 ,y1− x 2 ),(q2− p2 ,y2− x 2 ) )} = inf (y1,q1) {1 2 F∗T (2q1,2y1)+ 1 2 FT (2x−2y1,2p−2q1)+ + 4a−1 a ‖y1− x2‖ a+ 4b−1 b ‖q1− p2‖ b}, which, denoting (x1, p1) = (2y1,2q1) and (x2, p2) = (2x− 2y1,2p− 2q1), finally returns ( LaT )∗(p,x) = LaT (x, p), which establishes self-duality. LaT (x, p)≥ 〈x, p〉 comes from self-duality and the Fenchel-Young inequality, as in (3.1). Given proposition 2.17, and the definition of LaT , it is clear that L a T (x, p) = 〈x, p〉 ⇐⇒ FT (x, p) = F∗T (p,x) = 〈x, p〉 ⇐⇒ (x, p) ∈ T .  Remark 3.7 Compare (3.3) with (2.5) and (2.2). T can be parametrized by T = {(x, p) ∈ X×X∗ : LaT (x, p) = 〈x, p〉}, (3.4) 24 in an analogous fashion as in (2.3) and (2.6). The above listed results show that every maximal monotone operator has a self- dual Lagrangian that realizes it. Conversely, to any self-dual Lagrangian there is a maximal monotone operator being represented by it. It must be noted that the self-dual Lagrangian representation of a maximal monotone operator is not unique (see also [BWY10] where specific examples are provided), however the maximal monotone operator corresponding to a self-dual Lagrangian is unique. Example 3.8 Two very illustrative and important examples of self-dual Lagrangians follow: 1. For Φ ∈ Γ(X), L(x, p) :=Φ(x)+Φ∗(p) is a self-dual Lagrangian and ∂L= ∂Φ. 2. Let now S : X → X∗ be a linear continuous skew adjoint operator; L(x, p) := Φ(x)+Φ∗(p−Sx) is a self-dual Lagrangian and ∂L= ∂Φ+S. 3.5 ε-approximation of maximal monotone operators As shown, inequality (3.1) can be deduced from self-duality directly from the orig- inal Fenchel-Young inequality. Going back to the definition of an ε-subdifferential, this allows for a natural way to extend that concept to maximal monotone opera- tors: Given T a maximal monotone operator and its corresponding potential, LT ∈ L (X), for a parameter ε > 0 we can define an ε-approximant operator Tε the following way: Tεx := {p ∈ X∗ : LT (x, p)−〈x, p〉 ≤ ε}. The importance of this notion lies in the following natural extension of the Brøndsted-Rockafellar Lemma: 25 Theorem 3.9 Let L : X ×X∗→ R∪{+∞} be a self-dual Lagrangian and assume that for a pair (x0, p0) ∈ X×X∗, we have L(x0, p0)−〈x0, p0〉 ≤ ε. Then, for any λ > 0, there exists a pair (xλε , pλε ) ∈ ∂L such that 1. ‖xλε − x0‖ ≤ √ ε λ , 2. ‖pλε − p0‖ ≤ λ √ ε, Proof: First assume that M is a self-dual Lagrangian such that M(0,0) ≤ ε . We claim that there exists then a pair (yλε ,q λ ε ) ∈ ∂M such that 1. ‖yλε ‖ ≤ √ ε λ , 2. ‖qλε ‖ ≤ λ √ ε , Indeed, consider J, the duality mapping from X to X∗ and use the fact that ∂M is a maximal monotone operator and the characterization given by Theorem 2.9 to find x̃ ∈ X such that −λ 2Jx̃ ∈ ∂M(x̃). It follows that M(x̃,−λ 2Jx̃) = 〈x̃,−λ 2Jx̃〉=−λ 2‖x̃‖2. Now, since M is self-dual, we have M(0,0) =M∗(0,0) = sup (x,p)∈X×X∗ −M(x, p)≥−M(x̃,−λ 2Jx̃) = λ 2‖x̃‖2, from which we obtain that ‖x̃‖2 ≤ ελ 2 . Since ‖x̃‖ = ‖Jx̃‖, it suffices to set yλε := x̃ and qλε := λ 2Jx̃, to obtain that ‖qλε ‖= λ 2‖yλε ‖ ≤ λ √ ε . To complete the proof, we apply the above arguments to M(x, p) := L(x+ x0, p+ p0)−〈x, p0〉−〈x0, p〉−〈x0, p0〉, which is a self-dual Lagrangian on X×X∗. The hypothesis yields that M(0,0) = L(x0, p0)−〈x0, p0〉 ≤ ε. 26 It then follows from the above that there exists a pair (yε ,qε) ∈ ∂M such that ‖yλε ‖≤ √ ε λ , ‖qλε ‖≤ λ √ ε . For xλε := yλε +x0 and pλε := qλε + p0,we have L(xλε , pλε )= 〈xλε , pλε 〉, and therefore (xλε , pλε ) ∈ ∂L. Note also that ‖xλε − x0‖ ≤ √ ε λ and ‖pλε − p0‖∗ ≤ λ √ ε .  Remark 3.10 In view of item 1 in example 3.8, the above result contains the orig- inal form of the Brøndsted-Rockafellar Lemma. Remark 3.11 The notion of an ε-approximation of a maximal monotone operator depends on an already given choice of self-dual representation. Given the analo- gous properties of Fitzpatrick’s function, it should be clear that this notion could also be defined by using inequality FT (x, p)−〈x, p〉 ≤ ε. These notions are not necessarily equivalent, but the analogue Brøndsted-Rockafellar property corresponding to this notion holds, the proof (which is similar as the given above) can be found in [BS99], where Burachik & Svaiter introduce the concept this way, under the name of ε-enlargments. For clarity, this result is stated next. Theorem 3.12 Let T : X⇒ X∗ be a maximal monotone operator and let FT denote its corresponding Fitzpatrick function; assume that for a pair (x0, p0)∈X×X∗ and some ε > 0, we have FT (x0, p0)−〈x0, p0〉 ≤ ε. Then, for any λ > 0, there exists a pair (xλε , pλε ) ∈ T such that 1. ‖xλε − x0‖ ≤ √ ε λ , 2. ‖pλε − p0‖ ≤ λ √ ε, 3.6 A variational principle As seen above, a maximal monotone operator T can be written as T = ∂̄L where L is a selfdual Lagrangian on X×X∗, in such a way that solving for x in the equation p ∈ T (x), (3.5) 27 amounts to minimizing over x the non-negative functional (see (3.1)) I(x) := L(x, p)−〈x, p〉, and showing this minimum is actually zero (see proposition 3.3). This functional will play a fundamental role later in this work. The following result originally established in [Gho07] (see also [Gho08a]), gives sufficient conditions for the infimum of self-dual Lagrangians to be attained, and (as importantly) to be zero. Theorem 3.13 Let L be a self-dual functional on a reflexive Banach space X × X∗ such that for some x0 ∈ X, the functional q→ L(x0,q) is bounded above on a neighborhood of the origin in X∗. Then there exists x̄ ∈ X such that I(x̄) = min x∈X I(x) = 0. Remark 3.14 An important aspect of the above results is that they give conditions that yield that the actual value of the infimum is zero! This allows to use the variational principle for equations for which the existence of solutions is not a priori known. Self-duality plays a role in this part of the results and this was one of the original motivations to choose self-duality as an important property of the convex potentials. Remark 3.15 The boundedness condition on Theorem 3.13 is usually obtained by a coersiveness condition and self-duality: For example, if for some constant C1 we have C1(1−‖x‖2+ ‖p‖2) ≤ L(x, p), then by conjugation we obtain that for some constant C2 we have L∗(x, p)≤C2(1+‖x‖2+‖p‖2). 3.7 Coercivity conditions Coercivity conditions are often imposed when working with variational techniques. Usually they provide the needed compactness. In this section we will show how the choice LaT is appropriate in the sense that it preserves coercivity conditions from those of T . Let us begin by defining growth and boundedness conditions for monotone op- erators. As an example, an intuitive way to impose super linear growth conditions 28 on T would be by requiring that for some constantC > 0 C(‖x‖2−1)≤ 〈x, p〉, ∀(x, p) ∈ T. (3.6) On the other hand, if one wishes to impose sublinear growth conditions, a direct way would be to ask, for some C̃ > 0 ‖p‖ ≤ C̃(‖x‖+1), ∀(x, p) ∈ T. (3.7) Observe that the above condition can be imposed in an alternative way: If one requires, for some D > 0, that for any (x, p) ∈ T , one has D(‖p‖2− 1) ≤ 〈x, p〉, then, since 〈x, p〉 ≤ ‖x‖‖p‖, ‖p‖ must be bounded as in (3.7). Hence conditions (3.6) and (3.7) can be summarized in the following condition: (x, p) ∈ T ⇒max{C(‖x‖2−1), D(‖p‖2−1)} ≤ 〈x, p〉. As a slight generalization of the above, convenient for the setting of Lp spaces, the standard form of growth conditions for monotone operators will be given as follows: for a> 1, 1a + 1 b = 1, there exists constantsC,D> 0 such that (x, p) ∈ T ⇒max{C(‖x‖ a a −1), D(‖p‖ b b −1)} ≤ 〈x, p〉. (3.8) Proposition 3.16 A given maximal monotone operator T satisfies growth condi- tions (3.8) if and only if LaT satisfies, for some constants M,N > 0, M(‖x‖a+‖p‖b−1)≤ LaT (x, p)≤ N(‖x‖a+‖p‖b+1). (3.9) Proof: First let us assume that LaT obeys (3.9). For any (x, p) ∈ T we have that LaT (x, p) = 〈x, p〉, which from the left inequality in (3.9) gives that M(‖x‖a+‖p‖b−1)≤ 〈x, p〉, which for some constantsC and D gives (3.8). Conversely, assume that T satisfies (3.8). Consider FT , its Fitzpatrick’s func- 29 tion: FT (x, p) = sup (y,q)∈T {〈x,q〉+ 〈y, p〉−〈y,q〉} ≤ sup (y,q)∈T {〈x,q〉+ 〈y, p〉− C 2a (‖y‖a−1)− D 2b (‖q‖b−1)} ≤ ( C 2a (‖y‖a−1)+ D 2b (‖q‖b−1) )∗ (p,x) = 1 a ( D 2 )1−a ‖x‖a+ 1 b ( C 2 )1−b ‖p‖b+ C 2a + D 2b . Consider now (0, p0) ∈ T (some such p0 must exist from the boundedness con- ditions and maximality). From the definition of LaT , taking (x2, p2) = (0, p0) and (x1, p1) = (2x,2p− p0), LaT (x, p)≤ 1 2 FT (2x,2p− p0)+ 14a‖2x‖ a+ 1 4b ‖2p‖b, which using the bound obtained for FT gives LaT (x, p)≤ 1 a ( D 2 )1−a ‖2x‖a+ 1 b ( C 2 )1−b ‖2p−2p0‖b+ C2a+ D 2b + 1 4a ‖2x‖a+ 1 4b ‖2p‖b, which gives, for some constant N : LaT (x, p)≤ N(‖x‖a+‖p‖b+1). Taking the conjugate in the above inequality, since LaT is self-dual, yields the con- verse inequality.  The above connection between “standard growth conditions” does not neces- sarily follow for any self-dual potential, and in particular, Fitzpatrick’s functions fail to give a full connection of coercivity conditions as illustrated in the following example. Example 3.17 Choose X = R= X∗ and Tx= x. It is easy to compute FT (x, p) = (x+ p) 4 2 . 30 The above is not coercive in R2. On the other hand LT (x, p) = x 2 2 + p 2 2 . 3.8 Space dependent maximal monotone operators Time dependent monotone operators and self-dual Lagrangians and their “lifting” to appropriate Banach spaces of time dependent functions (path spaces) are treated in [Gho08a, Gho08b]. The case of space dependent monotone operators and space dependent Lagrangians appears in more detail in [GMZ10], fromwhere this section is taken. The class MΩ,p(RN) We introduce a particular class of space dependent operators on a given domain with specific growth conditions. This class appears in various works (see for ex- ample [PDD90]) and is a standard choice. The key inequality below reproduces (3.8). Definition 3.18 For a domain Ω in RN , p > 1 and 1p + 1 q = 1, we denote by MΩ,p(RN) the class of all possibly multi-valued functions T : Ω×RN → RN with closed values, which satisfy the following conditions: (i) T is measurable with respect to L (Ω)×B(RN) and B(RN) where L (Ω) is is the σ -field of all measurable subsets of Ω andB(RN) is the σ -field of all Borel subsets of RN . (ii) For a.e. x ∈Ω, the map T (x, .) : RN → RN is maximal monotone. (iii) There exist non-negative constants m1,m2,c1 and c2 such that for every ξ ∈RN and η ∈ T (ξ ), 〈ξ ,η〉RN ≥max { c1 p |ξ |p−m1, c2q |η | q−m2 } , (3.10) holds, where 〈., .〉RN is the inner product in RN . In this section, we establish a correspondence between maximal monotone maps in MΩ,p(RN) and a class of Ω-dependent self-dual Lagrangians. 31 Self-dual Lagrangians associated toMΩ,p(RN) Definition 3.19 Let Ω be a domain in RN . (i) A function L : Ω×RN ×RN → R∪{+∞} is said to be an Ω-dependent Lagrangian on Ω×RN×RN , if it is measurable with respect to the σ -field gener- ated by the products of Lebesgue sets in Ω and Borel sets in RN×RN . (ii) Such a Lagrangian L is said to be self-dual on Ω×RN ×RN if for any x ∈ Ω, the map Lx : (a,b)→ L(x,a,b) is a self-dual Lagrangian on RN ×RN , i.e., if L∗(x,b,a) = L(x,a,b) for all a,b ∈ RN where L∗(x,b,a) = sup{〈b,ξ 〉RN + 〈a,η〉RN −L(x,ξ ,η) : (ξ ,η) ∈ RN×RN}. The following was proved in [Gho08b] for a single maximal monotone operator. Proposition 3.20 If T ∈MΩ,p(RN) for some p> 1, then there exists anΩ-dependent self-dual Lagrangian L : Ω×RN ×RN → R such that T (x, .) = ∂̄L(x, .) for a.e. x ∈Ω and C0(|a|p+ |b|q−n0(x))≤ L(x,a,b)≤C1(|a|p+ |b|q+n1(x)) for all a,b ∈RN . (3.11) where C0 and C1 are two positive constants and n0,n1 ∈ L1(Ω). Conversely, if L : Ω×RN ×RN → R is an Ω-dependent self-dual Lagrangian sat- isfying (3.11), then ∂̄L(x, .) ∈MΩ,p(RN). The proof of the above contains arguments similar to the ones used through- out this chapter, adapted to the special case of space dependent operators and La- grangians. Proof. Let F : Ω×RN ×RN → R∪{+∞} be the Fitzpatrick function [Fit88] associated to T , i.e., F(x,a,b) := sup{〈b,ξ 〉RN + 〈a−ξ ,η〉RN ;η ∈ T (x,ξ )}. Note that measurability assumptions on T ensure that F is a normal integrand. Also, by the properties of the Fitzpatrick function [Gho08a], it follows that F∗(x,b,a)≥ F(x,a,b)≥ 〈a,b〉RN for a.e. x ∈Ω and for all a,b ∈ RN . 32 Moreover, η ∈ T (x,ξ ) if and only if F∗(x,η ,ξ ) = F(x,ξ ,η) = 〈η ,ξ 〉RN a.e. x ∈Ω. (3.12) Define L :Ω×RN×RN → R by L(x,a,b) = inf {1 2 F(x,a1,b1)+ 1 2 F∗(x,b2,a2)+ 1 4p |a1−a2|p+ 14q |b1−b2| q; (a,b) = 1 2 (a1,b1)+ 1 2 (a2,b2) } . We shall show that L is an Ω-dependent self-dual Lagrangian such that F∗(x,b,a)≥ L(x,a,b)≥ F(x,a,b) for a.e. x ∈Ω and for all a,b ∈ RN . (3.13) Fix a,b ∈ RN .We have L∗(x,b,a) = sup ξ ,η∈RN {〈ξ ,b〉RN + 〈a,η〉RN −L(x,ξ ,η)} = sup ξ ,η∈RN { 〈ξ ,b〉RN + 〈a,η〉RN − 1 2 F(x,ξ1,η1)− 12F ∗(x,ξ2,η2)+ − 1 4p |ξ1−ξ2|p− 14q |η1−η2| q ; (ξ ,η) = 1 2 (ξ1,η1)+ 1 2 (ξ2,η2) } = 1 2 sup ξ1,ξ2,η1,η2∈RN { 〈ξ1+ξ2,b〉RN + 〈a,η1+η2〉RN −F(x,ξ1,η1)+ −F∗(x,ξ2,η2)− 12p |ξ1−ξ2| p− 1 2q |η1−η2|q } . Using the fact that the Fenchel dual of the sum is the inf-convolution, we obtain L∗(x,b,a) = 1 2 inf a1,b1∈RN { F∗(x,b1,a1)+F(x,2a−a1,2b−b1)+ + 2q−1 q |b−b1|q+ 2 p−1 2p |a−a1|p } . 33 Setting a2 = 2a− a1 and b2 = 2b− b1 we have a = a1+a22 and b = b1+b22 . It then follows that L∗(x,b,a) = 1 2 inf a1,b1,a2,b2∈RN { F∗(x,b1,a1)+F(x,a2,b2)+ 2q−1 q |b1−b2 2 |q+ + 2p−1 2p |a1−a2 2 |p ; (a,b) = 1 2 (a1,b1)+ 1 2 (a2,b2) } = inf {1 2 F∗(x,b1,a1)+ 1 2 F(x,a2,b2)+ 1 4q |b1−b2|q+ + 1 4p |a1−a2|p ; (a,b) = 12(a1,b1)+ 1 2 (a2,b2) } = L(x,a,b). Thus, L is an Ω-dependent self-dual Lagrangian. Inequalities (3.13) simply follow from the definition and self-duality of L. We shall now prove that L satisfies the estimate (3.11). Note first that for all η ∈ T (x,ξ ) we have 1 p |ξ |p+ 1 q |η |p ≤ m1+m2+(c1+ c2)〈ξ ,η〉RN . It follows from the definition of the Fitzpatrick function F that F(x,a,b) = sup{〈b,ξ 〉RN + 〈a−ξ ,η〉RN ;η ∈ T (x,ξ )} ≤ sup { 〈b,ξ 〉RN + 〈a,η〉RN − 1 p(c1+ c2) |ξ |p− 1 q(c1+ c2) |η |q+ − m1+m2 c1+ c2 ;η ∈ T (x,ξ ) } ≤ sup ξ ,η∈RN { 〈b,ξ 〉RN + 〈a,η〉RN − 1 p(c1+ c2) |ξ |p+ − 1 q(c1+ c2) |η |q− m1+m2 c1+ c2 } = (c1+ c2)p−1 p |a|p+ (c1+ c2) q−1 q |b|q+ m1+m2 c1+ c2 . (3.14) Let η0(x)∈ T (x,0). By assumption |η0(x)|q ≤m2+〈0,η0(x)〉=m2 for a.e. x∈Ω, from which we get η0 ∈ Lq(Ω). It also follows from (3.12) that F∗(x,η0(x),0) = 0 34 for a.e. x ∈Ω. From the definition of L and (3.14), we get that L(x,a,b) ≤ 1 2 F(x,2a−η0(x),2b)+ 12F ∗(x,η0(x),0)+ 2q 4q |b|q+ 2 p 4p |a−η0(x)|p ≤ C1(|a|p+ |b|q+n1(x)) a.e. x ∈Ω, where C1 is a positive constant and n1 ∈ L1(Ω). The reverse inequality follows from the selfduality of L. Conversely, let L be a Ω-dependent self-dual Lagrangian satisfying (3.11). If η ∈ ∂̄L(x,ξ ) then 〈ξ ,η〉= L(x,ξ ,η)≥C0(|ξ |p+ |η |q−n0(x)), from which we conclude that ∂̄L(x, .) ∈MΩ,p(RN).  3.9 Self-dual Lagrangians onW 1,p0 (Ω)×W−1,q(Ω) We now show how one can “lift” an Ω-dependent self-dual Lagrangian to a self- dual Lagrangian on the phase spaceW 1,p0 (Ω)×W−1,q(Ω). This will allow us to give a variational formulation and resolution –via Theorem 3.13– of equations involving maximal monotone operators in divergence form. The following extends a result in [Gho08b]. Theorem 3.21 Let T ∈ MΩ,p(RN) for some p > 1, then for every w ∈W−1,q(Ω) with 1p + 1 q = 1, there exist ū ∈W 1,p0 (Ω) and f̄ (x) ∈ Lq(Ω;RN) such that{ f̄ ∈ T (x,∇ū(x)) a.e. x ∈Ω −div( f̄ ) = w. (3.15) It is obtained by minimizing the functional I(u) := inf f∈Lq(Ω;RN) −div( f )=w ∫ Ω [ L ( x,∇u(x), f (x) )−〈u(x), p(x)〉RN]dx on W 1,p(Ω), where L is an Ω-dependent self-dual Lagrangian on Ω×RN ×RN 35 associated to T in such a way that ∂̄L(x, ·) = T (x, ·) for a.e x ∈Ω. The above theorem will follow from the representation of a maximal monotone map inMΩ,p(RN) by anΩ-dependent self-dual Lagrangian onΩ×RN×RN (Propo- sition 3.20) combined with the following two propositions. Proposition 3.22 Suppose L is anΩ-dependent self-dual Lagrangian onΩ×RN× RN such that L(·,0,0)∈L1(Ω). Then the Lagrangian defined onW 1,p0 (Ω)×W−1,q(Ω) by F(u, p) := inf{ ∫ Ω L ( x,∇u(x), f (x) ) dx; f ∈ Lq(Ω;RN),−div( f ) = w}, (3.16) is selfdual. Proof: DenoteW 1,p0 (Ω) by X and its dualW −1,q(Ω) by X∗. For a fixed (v∗,v) ∈ X∗×X , we have F∗(v∗,v) = sup{〈u,v∗〉+ 〈p,v〉−F(u, p);u ∈ X , p ∈ X∗} = sup f∈Lq(Ω;RN) −div( f )=w (u,p)∈X×X∗ { 〈u,v∗〉+ 〈p,v〉− ∫ Ω L ( x,∇u(x), f (x) ) dx } = sup{〈u,v∗〉−〈div( f ),v〉− ∫ Ω L ( x,∇u(x), f (x) ) dx;u ∈ X , f ∈ Lq(Ω;RN)} = sup{〈u,v∗〉+ 〈 f ,∇v〉− ∫ Ω L ( x,∇u(x), f (x) ) dx;u ∈ X , f ∈ Lq(Ω;RN)}. Now set E := {g ∈ Lp(Ω;RN);g=∇u, u ∈ X} and let χE be the indicator function in Lp(Ω;RN), e.g., χE(g) = { 0 g ∈ E, +∞ elsewhere. An easy computation shows that χ∗E( f ) = { 0 div( f ) = 0, +∞ elsewhere. 36 Fix f0 ∈ Lq(Ω;RN) with −div( f0) = v∗. It follows that F∗(v∗,v) = sup{〈g, f0〉+ 〈 f ,∇v〉− ∫ Ω L ( x,g(x), f (x) ) dx−χE(g) ; g ∈ Lp(Ω;RN), f ∈ Lq(Ω;RN)} = inf{ ∫ Ω L∗(x, f0− f ,∇v)dx+χ∗E( f ); f ∈ Lq(Ω;RN)}. Here we have used the fact that (∫ ΩL ( x, ., . ) dx )∗(g, f ) = ∫ΩL∗(x, f (x),g(x))dx that holds since L(.,0,0) ∈ L1(Ω).We finally get F∗(v∗,v) = inf{ ∫ Ω L∗(x, f0− f ,∇v)dx; f ∈ Lq(Ω;RN),div( f ) = 0} = inf{ ∫ Ω L(x,∇v, f0− f )dx; f ∈ Lq(Ω;RN),div( f ) = 0} = inf{ ∫ Ω L(x,∇v, f )dx; f ∈ Lq(Ω;RN),−div( f ) = v∗} = F(v,v∗).  Here is our variational resolution for equation (3.15). Proposition 3.23 Suppose L is anΩ-dependent self-dual Lagrangian onΩ×RN× RN . Assume the following coercivity condition: L(x,a,b)≥ m(x)+C(|a|p+ |b|q) for all a,b ∈ RN , (3.17) where m ∈ L1(Ω) and C is a positive constant. Then for every w ∈W−1,q(Ω) the functional I(u) = inf f∈Lq(Ω;RN) −div( f )=w ∫ Ω [ L ( x,∇u(x), f (x) )−〈u(x),w(x)〉RN]dx attains its minimum at some ū ∈W 1,p0 (Ω) such that I(ū) = 0, and there exists f̄ ∈ Lq(Ω;RN) such that{ f̄ (x) ∈ ∂̄L(x,∇ū(x)) a.e. x ∈Ω −div( f̄ ) = w. 37 Proof. Take f0 ∈ Lq(Ω;RN) with −div ( f0(x) ) = w(x). Since L is an Ω-dependent self-dual Lagrangian,M(x,a,b) :=L(x,a,b+ f0(x))−〈a, f0(x)〉 is also anΩ-dependent self-dual Lagrangian on Ω×RN×RN . It follows from the above proposition that F(v,v∗) := inf f∈Lq(Ω;RN) −div( f )=v∗ ∫ Ω M ( x,∇v(x), f (x) ) dx is a self-dual Lagrangian onW 1,p0 (Ω)×W−1,q(Ω). In view of the coercivity condi- tion, Theorem 3.13 applies and there exists ū ∈W 1,p0 (Ω) such that F(ū,0) = inf f∈Lq(Ω;RN) −div( f )=0 ∫ Ω M ( x,∇ū(x), f (x) ) dx= 0. Using the coercivity condition again, we get that the above infimum is attained at some f1 ∈ Lq(Ω;RN) with div( f1) = 0. Thus, 0= F(ū,0) = ∫ Ω M ( x,∇ū(x), f1(x) ) dx = ∫ Ω [ L(x,∇ū(x), f1(x)+ f0(x))−〈∇ū(x), f0(x)〉RN ] dx = ∫ Ω [ L(x,∇ū(x), f1(x)+ f0(x))−〈∇ū(x), f1(x)+ f0(x)〉RN ] dx. Taking into consideration that L(x,∇ū(x), f1(x)+ f0(x))−〈∇ū(x), f1(x)+ f0(x)〉RN ≥ 0, we obtain that the latter is indeed zero, i.e., L(x,∇ū(x), f1(x)+ f0(x))−〈∇ū(x), f1(x)+ f0(x)〉RN = 0 for a.e. x ∈Ω. Setting f̄ := f1+ f0, we finally get that f̄ (x) ∈ ∂̄L(x,∇ū(x)) for a.e. x ∈Ω and that −div( f̄ ) = w.  38 Chapter 4 Topologies on spaces of convex Lagrangians and corresponding vector fields In this chapter we introduce and briefly study suitable topologies for the spaces of convex functions and of monotone operators. These topologies are given by the notions of Mosco convergence and Γ-convergence (Gamma convergence), for spaces of convex functions, and Kuratowski-Painlevé convergence for the space of maximal monotone operators. It will be shown that these convergence notions are related. Γ and Mosco convergence are now popular tools in the study of asymptotic properties of families of functionals. Γ-convergence, introduced by De Giorgi, is probably best known for its role in classic homogenization theory; on the other hand Mosco convergence is known to be well behaved under the Fenchel-Legendre transform. These notions are based on the more fundamental definition of set con- vergence known as Kuratowski-Painlevé. This mode of convergence has proven ideal for the study of convergence properties of families of maximal monotone op- erators as shown by the work of Kato, Brezis, Attouch and Browder, among others. 39 4.1 Mosco and Γ-convergence Mosco and Γ convergence are convergence notions for functions. These modes are not restricted to convex functions, but they are much better behaved in the context of proper, convex, lower semicontinuous functions, which is the natural setting we are working on. These modes of convergence are neither stronger nor weaker than pointwise convergence, they have a more geometrical nature and are shown to be ideal notions to use in variational settings. A good reference on the general subject is the book [Att84]. Definition 4.1 A sequence of functions { fn} ⊂ Γ(X) is said to Mosco converge to a function f , denoted fn M−→ f if both of the following conditions are satisfied: 1. For each weakly convergent sequence, xn ⇀ x, we have that f (x)≤ liminf n fn(xn). 2. For any x ∈ X, there exists a strongly convergent sequence xn → x such that f (x) = lim n fn(xn). The sequence { fn} is said to Γ-converge, denoted by fn Γ−→ f , if the above conditions hold, replacing the weak convergence in 1. by strong convergence. The above will also be denoted as f =M− lim fn and f = Γ− lim fn for Mosco and Γ convergence respectively. Conditions 1. and 2. are enough to guarantee uniqueness of the limits. Definition 4.2 A sequence satisfying condition 2. above will be referred to as a recovery sequence for { fn}. In terms of the epigraphs, this mode of convergence is that the set of limit points for the epigraphs {epi( fn)} is equivalent to its set of cluster points and this limit 40 corresponds to epi( f ), in both the strong and weak topologies (the strong topol- ogy). A fundamental property of Mosco convergence is the following: Lemma 4.3 fn M−→ f if and only if f ∗n M−→ f ∗. The above is not the case for Γ convergence (see the defined sequence in example 4.8). The following lemma (which appears in [AB93]) provides alternative defini- tions of Mosco convergence: Lemma 4.4 For fn, f ∈ Γ(X), the following are equivalent: 1. fn M−→ f . 2. The following two conditions hold: • For any x ∈ X, there exists a strongly convergent sequence, xn → x, such that f (x) = lim n fn(xn). • For any p ∈ X∗, there exists a strongly convergent sequence, pn → p, such that f ∗(p) = lim n f ∗n (pn). 3. The following two conditions hold: • For any x∈ dom(∂ f ), there exist a strongly convergent sequence, xn→ x, such that f (x)≥ limsup n fn(xn). • For any p∈ ran(∂ f ), there exists a strongly convergent sequence, pn→ p, such that f ∗(p)≥ limsup n f ∗n (pn). 41 Γ and Mosco convergence, as mentioned before, can be seen as Kuratowski- Painlevé convergence for epigraphs (in both the strong and weak topology in the case of Mosco convergence): 4.2 Kuratowski - Painlevé topology For a sequence of sets {An}, we use the norm topology to define Definition 4.5 1. The limit inferior set, LiAn, given by all limit points of the sequence {An}. In other words: x ∈ LiAn ⇐⇒ lim n ‖x−an‖= 0 for some sequence {an} with an ∈ An∀n. 2. The limit superior set, LsAn, given by all cluster points of the sequence {An}. In other words: x ∈ LsAn ⇐⇒ liminf n ‖x−an‖= 0 for some sequence {an} with an ∈ An∀n. 3. We say that the sequence converges to the set A in the sense of Kuratowski - Painlevé if A= LiAn = LsAn. This mode of convergence will be denoted by An K−P−−−→ A. This mode of convergence on the space X ×X∗ is ideal for a notion of conver- gence for maximal monotone operators, restricted to the case of convex subdiffer- entials it is known to match neatly with Mosco convergence, as it will be shown in the coming sections. We will also extend this relation, to some extent, to general maximal monotone operators and their corresponding convex potentials. G convergence, introduced by Dal Maso to study certain types of homoge- nization problems, is based on Kuratowski-Painlevé convergence (see for example [PDD90, Definition 3.5]). 42 4.3 Mosco convergence vs Γ-convergence for self-dual functions It is clear that Mosco convergence is a stronger notion of convergence than Γ- convergence, however, with the added condition of self-duality, these modes of convergence are actually equivalent. It is worth of mention that Γ-convergence in the general setting (without any restriction to self-dual functionals) combined with an equicoercivity condition is itself equivalent to Mosco convergence (that is: a family of equicoercive convex functions which is Γ-convergent will also be Mosco convergent). Theorem 4.6 Let {Ln} be a family of self-dual Lagrangians on X ×X∗, and let L be a function on X×X∗. The following statements are then equivalent: 1. {Ln} is Mosco convergent to L. 2. L is self-dual and {Ln} Γ-converges to L. 3. L is self-dual and for any (x, p) ∈ X ×X∗ there exists a sequence (xn, pn) converging strongly to (x, p) in X×X∗ such that limsup n Ln(xn, pn)≤ L(x, p). Proof: For (1)→ (2) we just need to prove that L is self-dual since Mosco convergence clearly implies Γ-convergence. Since L is the Mosco limit of Ln, it follows from Lemma 4.3 that L∗ is a Mosco limit of L∗n. Denoting LTn (p,x) := Ln(x, p) and L T (p,x) := L(x, p), it follows that LT is a Mosco-limit of LTn on X ∗×X . On the other hand, by self- duality of Ln we have that LTn = L ∗ n from which we obtain that L T =M− limnLTn = M− limnL∗n = L∗, and therefore LT = L∗, and L is therefore self-dual. (2)→(3) follows from the definition of Γ-convergence. For (3)→(1) we let (p,x) ∈ X∗×X and consider a sequence {(pn,xn)} ⊂ X∗×X such that (pn,xn)⇀ (p,x)weakly in X∗×X . By the definition of Fenchel-Legendre 43 duality we have liminf n L∗n(pn,xn) = liminfn sup(y,q)∈X×X∗ {〈xn,q〉+ 〈y, pn〉−Ln(y,q)}. (4.1) Consider now an arbitrary pair (x̃, p̃) and let {(x̃n, p̃n)} be the recovery sequence given by item (3.). It follows from (4.1) that liminf n L∗n(pn,xn)≥ liminfn ( 〈xn, p̃n〉+〈x̃n, pn〉−Ln(x̃n, p̃n) ) ≥〈x, p̃〉+〈x̃, p〉−L(x̃, p̃). Since (x̃, p̃) is arbitrary, taking the supremum over all (x̃, p̃) yields liminf n L∗n(pn,xn)≥ L∗(p,x). Since both Ln and L are self-dual, this implies that liminf n Ln(xn, pn)≥ L(x, p), and therefore that L is a Mosco-limit of Ln.  Remark 4.7 Note that while the Mosco convergence of self-dual Lagrangians au- tomatically implies that the limiting Lagrangian L is itself self-dual, this fails for Γ-convergence as shown in the following example. The following example is an adaptation of an counter example found in [DMS08]. Example 4.8 Let H be an infinite dimensional Hilbert space. Consider a set {en} with ‖en‖= 1 and en⇀ 0 (For example, the orthonormal basis of the space). Define Ln(x, p) := 1 2 ‖x− en‖2+ 12‖p‖ 2+ 〈p,en〉. Notice that Ln is self-dual. It can be checked directly that for any strongly conver- gent sequence (xn, pn)→ (x, p) in H×H we have limnLn(xn, pn) = L(x, p), where L(x, p) := 1 2 ‖x‖2+ 1 2 ‖p‖2+ 1 2 . This means that L is a Γ-limit of Ln. On the other hand, it is easily seen that 44 L∗(p,x) = L(x, p)− 1, so L is not self-dual and therefore we do not have Mosco convergence. 4.4 Kuratowski-Painlevé convergence for maximal monotone operators In a spirit similar to Theorem 4.6, where the definition for Mosco convergence can be simplified in the self-dual case, the following lemma is a standard result and provides a shorter definition of Kuratowski-Painlevé convergence for maximal monotone operators: Lemma 4.9 Let {Tn} be a sequence of maximal monotone operators and T a max- imal monotone operator. Then the following are equivalent: 1. T ⊂ LiTn. 2. Tn K−P−−−→ T . Proof: Assume that T ⊂ LiTn, we need to show that Ls(Tn) ⊂ T . Consider (y,q) ∈ Ls(Tn), thus, there exist some sequence (yn,qn) ∈ Tn such that for some subsequence (yn(k),qn(k)) → (y,q). Now take an arbitrary (x, p) ∈ T . From our assumption that T ⊂ LiTn, there exist a sequence (xn, pn) ∈ Tn such that (xn, pn)→ (x, p). For each k we have 〈xn(k)− yn(k), pn(k)−qn(k)〉 ≥ 0, and as k→ ∞ we get 〈x− y, p−q〉 ≥ 0. The above is for any (x, p) ∈ T , so by maximality, we must have that (y,q) ∈ T , and thus Ls(Tn)⊂ T . We conclude Tn K−P−−−→ T . The converse implication follows simply by definition.  45 4.5 Continuity of the regularization of self-dual Lagrangians The following well known result (see [Att84]) relates convergence of a sequence of convex functions with convergence of their Moreau-Yosida regularizations: Proposition 4.10 Let { fn} ⊂ Γ(X) and f ∈ Γ(X). The following are equivalent 1. fn M−→ f . 2. For some λ0 > 0, ( fn)λ0 M−→ ( f )λ0 . 3. For any λ > 0, ( fn)λ M−→ ( f )λ . In a similar spirit, it can be shown that some of the regularizations given in definition 2.21 also preserve Mosco convergence. Proposition 4.11 Let {Ln} and L be a sequence and an element of L (X). The following are equivalent 1. Ln M−→ L. 2. For some λ0 > 0, (Ln)λ01 M−→ (L)λ01 . 3. For any λ > 0, (Ln)λ1 M−→ (L)λ1 . 4. For some λ0 > 0, (Ln)λ02 M−→ (L)λ02 . 5. For any λ > 0, (Ln)λ2 M−→ (L)λ2 . The proof of the above will be split into various smaller results. Recalling definition 2.21, let us introduce the following: Definition 4.12 For any λ > 0 and any L ∈L (X), let RλL (x, p) := argmin y∈X { L(y, p)+ ‖x− y‖2 2λ + λ 2 ‖p‖2 } and T λL (x, p) := argmin q∈X∗ { L(x,q)+ ‖x‖2 2λ + λ 2 ‖p−q‖2 } . 46 Also, the following important boundedness property of Mosco convergence will be useful (a proof can be found in [DMS08]). Lemma 4.13 If {Ln} is a Mosco converging sequence of proper convex functionals on X×X∗, then there exist positive constants a and b such that Ln(x, p)+a(‖x‖+‖p‖)+b≥ 0 for all (x, p) ∈ X×X∗. We will begin by showing the following. Proposition 4.14 If L ∈L (X) is the Mosco-limit for some sequence {Ln} of self- dual Lagrangians: 1. For every λ > 0 Lλ1 =M− lim ( Ln )λ 1 and L λ 2 =M− lim ( Ln )λ 2 . 2a. For any converging sequence on X, xn → x, and any p ∈ X∗, there exist a converging sequence on X∗, pn → p such that:( Ln )λ 1 (xn, pn)→ Lλ1 (x, p). 2b. For any converging sequence on X∗, pn → p, and any x ∈ X, there exist a converging sequence on X, xn → x such that:( Ln )λ 2 (xn, pn)→ Lλ2 (x, p). 3. For any converging sequence (xn, pn)→ (x, p): RλLn(xn, pn)→ RλL (x, p) and T λLn(xn, pn)→ T λL (x, p). Proof: Consider any (x, p) ∈ X × X∗ and any converging sequence on X , xn → x. Denote y := RλL (x, p). Let (yn, pn)→ (y, p) be a corresponding recovery 47 sequence for {Ln}, that is: Ln(yn, pn)→ L(y, p). We have ( Ln )λ 1 (xn, pn)≤ Ln(yn, pn)+ ‖xn− yn‖2 2λ + λ 2 ‖pn‖2, and thus, since all the sequences {xn}, {yn} and {pn} are strongly convergent: limsup n ( Ln )λ 1 (xn, pn)≤ L(y, p)+ ‖x− y‖2 2λ + λ 2 ‖p‖2. Since y= RλL (x, p), the above is: limsup n ( Ln )λ 1 (xn, pn)≤ Lλ1 (x, p). In view of the third characterization for Mosco convergence given in Theorem 4.6 and the self-duality of ( Ln )λ 1 and L λ 1 (proposition 3.4), this inequality establishes the first equation in item 1: Lλ1 =M(X×X∗)− lim ( Ln )λ 1 . This in turn grants, by the liminf property of Mosco convergence, that for the sequence shown above: ( Ln )λ 1 (xn, pn)→ Lλ1 (x, p). This is conclusion 2a. To complete this proof, we turn to show that for any converging sequence (xn, pn)→ (x, p) such that ( Ln )λ 1 (xn, pn)→ Lλ1 (x, p), we have RλLn(xn, pn)→ RλL (x, p). 48 Considering such (xn, pn), define now yn := RλLn(xn, pn). We have ( Ln )λ 1 (xn, pn) = Ln(yn, pn)+ ‖xn− yn‖2 2λ + λ 2 ‖pn‖2. (4.2) Using the uniform boundedness from below for Mosco convergence (see lemma 4.13), since we have that {Ln} is a Mosco converging sequence, we have, for some positive constants a and b: ( Ln )λ 1 (xn, pn)≥ ‖xn− yn‖2 2λ + λ 2 ‖pn‖2−a(‖yn‖+‖pn‖)−b. Since the left hand side of the above inequality is a converging sequence, and so is {(xn, pn)}, then the sequence {yn}must be bounded. Let us then consider a weakly convergent subsequence which we will not relabel for simplicity: yn ⇀ ỹ. Take the lower limit on (4.2) to get Lλ1 (x, p)≥ liminfn Ln(yn, pn)+ liminfn ‖xn− yn‖2 2λ + liminf n λ 2 ‖pn‖2, which from Mosco convergence of {Ln} and lower semicontinuity of the norm yields Lλ1 (x, p)≥ L(ỹ, p)+ ‖x− ỹ‖2 2λ + λ 2 ‖p‖2; The above must then, be an equality and this gives: ỹ= RλL (x, p). Furthermore, we can assume, without loss of generality, that the subsequence was taken in such a way that Ln(yn, pn)→ liminfnLn(yn, pn), which gives that also ‖x− yn‖2 2λ → ‖x− ỹ‖ 2 2λ , and hence ‖yn‖→‖ỹ‖, so that subsequence actually converges strongly to RλL (x, p). This argument can be done for any subsequence of the original sequence {yn}, and hence we have shown the desired result, namely, RλLn(xn, pn)→ RλL (x, p). 49 This is the first part of conclusion 3. The proofs for the related results for ( Ln )λ 2 , Lλ2 and T λ L are directly analogous to the above.  Proposition 4.15 If for some λ0 > 0, we have one of Lλ01 = M− lim ( Ln )λ0 1 or Lλ02 =M− lim ( Ln )λ0 2 , then L=M− limLn. Proof: Consider (x, p) ∈ X × X∗. Assume that (x, p) ∈ dom(∂L). Take q ∈ ∂1L(x, p) (y ∈ ∂2L(x, p)). Define, for some ε > 0, λ = λ0+ ε (λ = λ0 ? ε). Define y := x−λJ−1q (q := p−λJy). It can be checked that x= RλL (y, p) (p= T λ L (x,q)). From proposition 2.23, we can apply proposition 4.13 to get that there exists a sequence (yn, pn)→ (y, p) such that( Ln )λ 1 (yn, pn)→ ( L )λ 1 (y, p) ( a sequence (xn,qn)→ (x,q) such that ( Ln )λ 2 (xn,qn)→( L )λ 2 (x,q) ) and RλLn(yn, pn)→ RλL (y, p) ( T λLn(xn,qn)→ T λL (x,q) ) . Define xn := RλLn(yn, pn) ( pn := T λLn(xn,qn) ) . Simply write now ( Ln )λ 1 (yn, pn) = Ln(xn, pn)+ ‖xn− yn‖2 2λ + λ 2 ‖pn‖2, (( Ln )λ 2 (xn,qn) = Ln(yn,qn)+ ‖xn‖2 2λ + λ 2 ‖pn−qn‖2 ) . Then lim n Ln(xn, pn) = ( L )λ 1 (y, p)− ‖x− y‖2 2λ − λ 2 ‖p‖2, ( lim n Ln(yn,qn) = ( L )λ 2 (x,q)− ‖x‖2 2λ − λ 2 ‖p−q‖2 ) , this is Ln(xn, pn)→ L(x, p) and, since Ln and L are self-dual, this is enough to guarantee L=M− limLn. 50 Corollary 4.16 Given Ln,L ∈L (X) and λ > 0, L=M− limLn ⇐⇒ ( L )λ 1 =M− lim ( Ln )λ 1 ⇐⇒ ( L )λ 2 =M− lim ( Ln )λ 2 . 4.6 Continuity of regularizations of maximal monotone operators Results analogous to Proposition 4.11 are already known for maximal monotone operators. The following can be found in [Att84]. Proposition 4.17 Let {Tn} be a sequence of maximal monotone operators and let T be a maximal monotone operator. The following are equivalent: 1. Tn K−P−−−→ T . 2. For some λ0 > 0, (Tn)λ0 K−P−−−→ Tλ0 . 3. For each λ > 0, (Tn)λ K−P−−−→ Tλ . 4. For some λ0 > 0, RTnλ0 K−P−−−→ RTλ0 . 5. For each λ > 0, RTnλ K−P−−−→ RTλ . 4.7 Attouch’s theorem: on the bicontinuity of the application f → ∂ f For each f ∈ Γ(X), define ∆( f ) := {(x, f (x), p) | (x, p) ∈ ∂ f}. Recall Attouch’s Theorem, it appears in [Att84] and [AB93]: Theorem 4.18 Let X be a reflexive Banach space and f , f1, f2, . . . proper convex functions on X. The following are equivalent: 51 1. fn M−→ f . 2. ∂ fn K−P−−−→ ∂ f and ∆( f )∩Li∆( fn) 6= /0. Essentially, this result establishes the equivalence of Mosco convergence of proper convex functions on X (equivalently, their Fenchel-Legendre conjugates on X∗) and Kuratowski - Painlevé convergence on X ×X∗ of their subdifferentials. The condition on ∆( f ) is just a normalizing condition, since a convex functional is uniquely defined by its subdifferential mapping only up to an additive constant. Remark 4.19 As a trivial example of the role of the condition on ∆( f ), consider f a fixed convex function in Γ(X) and the sequence given by fn(x) := f (x)+n. Clearly fn does not Mosco converge to f , but ∂ fn remains constant (and thus converges in the sense of Kuratowski-Painlevé). Observe we have f ∗n (p) = f ∗(p)−n, and hence fn+ f ∗n = f + f ∗. Observe that the problem lies in the additive constant that preserves the subdiffer- ential. This very simple idea will be used later, when we present a “self-dual” formu- lation of Attouch’s Theorem on phase space. Proofs of Attouch’s Theorem can be found on [Att84] and [AB93]. However, we present here the proof for convenience of the reader and since its arguments are of interest to us. Remark 4.20 Observe that the proof relies on two key ingredients: The Brønsted - Rockafellar Lemma, and Rockafellar’s integration formula. Proof of Theorem 4.18: Assume first that fn M−→ f . Fix (x, p) ∈ ∂ f , there exists, by the presented characterizations of Mosco convergence on lemma 4.4, a 52 sequence xn → x and pn → p, each converging strongly in X and X∗ respectively, such that fn(xn)→ f (x) and f ∗n (pn)→ f ∗(p). We have then f (x) + f ∗(p) = 〈x, p〉 = limn〈xn, pn〉, and hence, if we define εn := fn(xn)+ f ∗n (pn)−〈xn, pn〉, we obtain that limn εn = 0 and that pn is an εn- subdifferential of fn at xn. Hence, by Theorem 2.2, we have the existence of a pair (x̃n, p̃n)∈ ∂ fn such that ‖xn− x̃n‖<√εn and ‖pn− p̃n‖<√εn. Clearly, since εn → 0: x̃n → x and p̃n → p. This shows that ∂ f ⊂ Li(∂ fn). Now, since we have fn M−→ f , this gives that f (x)≤ liminf n fn(x̃n) (4.3) and f ∗(p)≤ liminf n f ∗n (p̃n), and since lim n ( fn(x̃n)+ f ∗n (p̃n) ) = lim n ( 〈x̃n, p̃n〉 ) = 〈x, p〉= f (x)+ f ∗(p), we can write now limsup n fn(x̃n)≤ limsup n ( fn(x̃n)+ f ∗n (p̃n) ) + limsup n ( − f ∗n (p̃n) ) = = f (x)+ f ∗(p)− liminf n f ∗n (p̃n)≤ f (x). In conjunction with (4.3), this shows lim fn(x̃n) = f (x). We have shown (x̃n, fn(x̃n), p̃n)→ (x, f (x), p), where (x̃n, p̃n)∈ ∂ fn. This shows that ∆( f )∩Li∆( fn) 6= /0. To finalize this part of the proof, we just need to show that Ls(∂ fn)⊂ ∂ f . But this follows from the already established fact ∂ f ⊂ Li(∂ fn), the maximal mono- tonicity of convex subdifferentials, and Lemma 4.9. 53 Now we establish the converse implication. Assume that ∂ fn K−P−−−→ ∂ f and that ∆( f )∩Li∆( fn) 6= /0. Fix some (y,q) ∈ ∂ f . From the assumption, we have a sequence (yn,qn) ∈ ∂ fn converging to (y,q). Pick now (x, p) such that (x, f (x), p) ∈ ∆( f )∩Li∆( fn). Let this condition be realized by the sequence (xn, pn) ∈ ∂ fn. Choose an arbitrary chain from y to x, that is, some finite set {xi}ki=1 with x1 = y and xk = x. Pick corresponding pi ∈ ∂ f (xi), with p1 = q and pk = p. Since ∂ fn K−P−−−→ ∂ f , we can consider, also, sequences {(xin, pin)}n with (xin, pin)∈ ∂ fn and (xin, pin)→ (xi, pi), where we require (xkn, pkn) = (xn, pn), with (xn, pn) as given above. Write now fn(xn)− fn(yn) = k−1 ∑ i=1 ( fn(xi+1n )− fn(xin) ) . From the subgradient inequality, fn(xn)− fn(yn)≥ k−1 ∑ i=1 〈xi+1n − xin, pin〉. Now, since we chose xn in such a way that limn fn(xn)= f (x), we apply limsupn to the above to get f (x)≥ limsup n fn(yn)+ k−1 ∑ i=1 〈xi+1− xi, pi〉, and, taking the sup over all chains, Rockafellar’s formula (see Theorem 2.14) yields f (x)≥ limsup n fn(yn)+ f (x)− f (y), which finally gives us the first condition in item 3. of Lemma 4.4: f (y)≥ limsup n fn(yn). 54 In an analogous way, using ∆( f ∗) = {(p, f ∗(p),x) : (x, p)∈ ∂ f}, we can show the second condition in item 3. of Lemma 4.4, namely, f ∗(p)≥ limsup n f ∗n (pn). This completes the proof.  4.7.1 Attouch’s theorem on phase space We propose the following reformulation of Attouch’s Theorem as a first step to- wards a generalization of the result for maximal monotone operators and self-dual Lagrangians. Note that the need of a normalizing condition is gone and the result in this form directly relates Mosco convergence (in phase space) to Kuratowski- Painlevé convergence: Theorem 4.21 Let X be a reflexive Banach space and f , f1, f2, . . . functions in Γ(X). The following are equivalent: 1. fn+ f ∗n M−→ f + f ∗. 2. ∂ fn K−P−−−→ ∂ f . Proof: The proof that the first condition implies the second condition is iden- tical to the proof used for Theorem 4.18. Assume, then, that ∂ fn K−P−−−→ ∂ f . Fix any (x0, p0) ∈ ∂ f and any sequence (xn, pn)→ (x0, p0) with (xn, pn) ∈ ∂ fn. Now define gn(x) := fn(x)− fn(xn) and g(x) := f (x)− f (x0). Observe that ∂gn = ∂ fn and ∂g = ∂ f , which, given the current assumption, implies ∂gn K−P−−−→ ∂g. By construction, we also have gn(xn) = g(x0) = 0, which yields (xn,gn(xn), pn)→ (x0,g(x0), p0), and thus ∆(g)∩Li∆(gn) 6= /0 so we can use Theorem 4.18 to obtain right away that, on X , gn M−→ g which in turn yields that also, on X∗, g∗n M−→ g∗. 55 Clearly this implies, in X×X∗, gn+g∗n M−→ g+g∗, and since g∗n(p) = f ∗n (p)+ fn(xn) and g∗(p) = f ∗(p)+ f (x0), we have gn+ g∗n = fn+ f ∗n and g+g∗ = f + f ∗, that is: fn+ f ∗n M−→ f + f ∗ which completes the proof.  4.8 L 7→ ∂L and FT 7→ T are continuous Our first result shows the continuity of the application L 7→ ∂L, as well as its ana- logue for Fitzpatrick’s functions. Theorem 4.22 Let X be a reflexive Banach space and suppose {Ln} is a family of selfdual Lagrangians on X ×X∗. If L : X ×X∗ → R∪{+∞} is a selfdual La- grangian that is a Mosco limit of {Ln}, then the graph of ∂̄Ln converge to the graphs of ∂̄L in the sense of Kuratowski-Painlevé. The following is the corresponding analogue for Fitzpatrick’s functions. Its proof is directly analogous to the one for the previous result, using the Brønsted- Rockafellar property of the Fitzpatrick function (see Remark 3.11), and will not be specifically written. Theorem 4.23 Let X be a reflexive Banach space and suppose that T and the ele- ments of a family {Tn} are maximal monotone operators on X. If FT , the Fitzpatrick function for T , is the Mosco-limit of {FTn}, where FTn is the Fitzpatrick function for {Tn}, then {Tn} converges to T in the sense of Kuratowski-Painlevé. Proof of Theorem 4.22. Fix (x, p) ∈ ∂L. Then in view of the Mosco conver- gence, there exists a sequence (xn, pn) converging strongly to (x, p) in X×X∗ such that Ln(xn, pn)→ L(x, p). We then have L(x, p) = 〈x, p〉= limn〈xn, pn〉, and there- fore if we define εn := Ln(xn, pn)−〈xn, pn〉, we obtain that limn εn = 0. Hence, by Lemma 3.9, we have the existence of a pair (x̃n, x̃∗n)∈ ∂Ln such that ‖xn− x̃n‖< √ εn 56 and ‖pn− x̃∗n‖∗ < √ εn. Clearly x̃n → x and x̃∗n → p as εn → 0. This shows that ∂L⊂ Li(∂Ln), and the proof is complete, in view of Lemma 4.9.  4.9 On the continuity of ∂L 7→ L Before attempting a converse result, namely, stating the continuity of the appli- cation ∂L 7→ L, it is important to be aware that, as mentioned before, self-dual Lagrangians are not unique, and hence we cannot expect any perfect analogy to Theorem 4.21. The first obvious ingredient we are missing is an integration formula for a given maximal monotone operator T . A direct substitute for it is the selection of LaT , which is uniquely determined. Indeed the following result can be shown: Theorem 4.24 Let {Tn} be a sequence of maximal monotone operators, and let T be a maximal monotone operator. If the sequence {FTn} Mosco converges to FT , then, for each fixed a> 1, we have that LaTn M−→ LaT . The proof follows immediately from the general fact below: Proposition 4.25 Let {Fn} be a sequence of functions in Γ(X ×X∗) Mosco con- vergent to F. For any constants A,B and a,b > 1, we have that the sequence of functions {Gn}, defined by Gn(x, p) = inf (y,q) {Fn(y,q)+F∗n (2p−q,2x− y)+A‖x− y‖a+B‖p−q‖b}, Mosco converges to the function G, given by G(x, p) = inf (y,q) {F(y,q)+F∗(2p−q,2x− y)+A‖x− y‖a+B‖p−q‖b}. Proof: Fix (x, p). There exists some (y,q) such that G(x, p) = F(y,q)+F∗(2p−q,2x− y)+A‖x− y‖a+B‖p−q‖b. Observe that since {Fn} is Mosco convergent, so is {F∗n }. Define z := 2x− y and r := 2p−q. There exist sequences (yn,qn) and (zn,rn), strongly convergent to (y,q) 57 and (z,r) respectively, and such that lim n Fn(yn,qn) = F(y,q) and lim n F∗n (rn,zn) = F ∗(r,z). Define xn := yn+zn2 and pn := qn+rn 2 . It can be checked that (xn, pn) is convergent to (x, p). Also Gn(xn, pn) ≤ Fn(yn,qn)+F∗n (2pn−qn,2xn− yn)+A‖xn− yn‖p+B‖pn−qn‖q = Fn(yn,qn)+F∗n (rn,zn)+A‖xn− yn‖p+B‖pn−qn‖q. Taking the upper limit yields limsup n Gn(xn, pn)≤ F(y,q)+F∗(r,z)+A‖x− y‖p+B‖p−q‖q = G(x, p). The above can be repeated for G∗, yielding the result.  However, whether or not the convergence of {Tn} to T implies the convergence of their respective Fitzpatrick’s functions, remains without proof, to this point. 4.10 A compact class in Lp In the following uwill denote a function in Lp(Ω), 1< p<∞, whereΩ is a domain on Rn. For q, the conjugate exponent of p (i.e. such that 1p + 1 q = 1), v will denote a function in Lq(Ω). We will consider a class of functionals of the form F = {F(u) = ∫ Ω f ( u(x) ) dx : f ∈ F̃}, where the class F̃ ⊂ Γ(R) is specified by constants βi,γi, i= 1,2: F̃ = { f : R→ R : β1|x|p− γ1 < f (x)< β2|x|p+ γ2 and f ∈ Γ(R)}. 58 Proposition 4.26 The classF is Γ compact in Lp(Ω). Proof: Consider a sequence {Fn} inF ; by taking Fn(u) = ∫ Ω fn(u(x))dx, this sequence corresponds in a natural way to a sequence { fn} in F̃ . Since the class F̃ consists entirely of convex functions, we can show they are also equicontinuous on compact subsets of R: consider v(x) ∈ ∂ f (x), we have for any h ∈ R 〈v(x),h〉 ≤ f (x+h)− f (x), taking the sup over {|h| = 1} and considering the definition of the class, we get that there must be a constantC such that |v(x)| ≤C(|x|p+1). This gives that the subdifferentials have uniformly bounded norms on compact subsets ofR, which implies equicontinuity on compact subsets ofR. By the Arzela- Ascoli lemma, for any compact K ⊂ R, there exists a subsequence that converges uniformly on K to some fK ∈ F̃ . By covering R with an increasing compact cover, we can now construct an appropriate subsequence (denoted by the same symbol) { fn} such that fn(x)→ f∞(x) for each x ∈ R and uniformly on compact subsets of R. Clearly f∞(x) ∈ F̃ . Consider now any subsequence {un} in Lp(Ω) converging in norm to u. Focus on an a.e. converging subsequence, again denoted by {un}. By simply applying Fatou’s lemma we get∫ Ω f∞ ( u(x) ) dx≤ liminf ∫ Ω fn ( un(x) ) dx which, taking F∞(u) = ∫ Ω f∞ ( u(x) ) dx, gives F∞(u)≤ liminfFn(un). Consider now any u ∈ Lp(Ω). Since fn → f pointwise, and again, considering 59 the bounds defining the class, by the dominated convergence theorem we get that Fn(u) = ∫ Ω fn ( u(x) ) dx→ ∫ Ω f∞ ( u(x) ) dx= F∞(u). We have shown that Fn Γ−→ F∞.  We can also define the class F ∗ := {F∗ : F ∈F}. It is the case that F ∗ = {G(v) = ∫ Ω f ∗ ( v(x) ) dx : f ∈ F̃}, A direct computation can show that if f ∈ F̃ , then( β2|x|p− γ2 )∗ ≥ f ∗(x)≥ ( β1|x|p− γ1 )∗ . Corollary 4.27 The classF ∗ is Γ compact. In order to show that the class F has the liminf property also for weakly convergent sequences, Barbu and Kunisch used the following lemma (see [BK95, BK96]): Lemma 4.28 Consider a sequence of functions { fn} ⊂ F̃ converging to f0 ∈ F̃ uniformly on compact sets of R. Let v ∈ Lp(Ω). Then (1+λ∂ fn)−1v→ (1+λ∂ f0)−1v strongly on Lp(Ω). Consequently there is a subsequence { fnk} such that (1+λ∂ fnk) −1v(x)→ (1+λ∂ f0)−1v(x) for a.e. x ∈Ω. We can work without this result by using the properties of Mosco convergence instead. (Of course, our proofs used similar machinery as the lemma above). 60 Corollary 4.29 The classF is Mosco compact. Proof: From Proposition 4.26 and corollary 4.27, for any sequence {Fn}⊂F there exists a subsequence, denoted by the same symbol, such that for any u ∈ Lp(Ω) lim n Fn(u) = F(u) and for any v ∈ Lq(Ω) lim n F∗n (v) = F ∗(v). Referring to item 2 in lemma 4.4, we have that Fn M−→ F.  The following simple lemma will be useful in the sequel: Lemma 4.30 If {Fn} ⊂F is such that Fn M−→ F, then: 1. For any strongly convergent sequence in X∗, pn → p, F∗n (pn)→ F∗(p). 2. For any strongly convergent sequence in X, xn → x, Fn(xn)→ F(x). Proof: Consider a sequence pn → p. Define xn to be an element such that 〈xn, pn〉−Fn(xn) = F∗n (pn). Such xn exists thanks to the boundedness conditions on F , and for the same con- ditions we must have that the sequence {xn} remains bounded in X . Hence we can extract a weakly convergent subsequence, again denoted by the same symbol, xn ⇀ x. We have for this subsequence limsup n F∗(pn)≤ 〈x, p〉− liminf n Fn(xn)≤ 〈x, p〉−F(x)≤ F∗(p). 61 Since we also have that liminfnF∗n (pn)≥ F∗(p), we obtain the desired conclusion, that F∗n (pn)→ F∗(p). Observing the analogue bound conditions forF ∗ and that Fn M−→ F if and only if F∗n M−→ F∗, the same argument as above yields Fn(xn)→ F(x) for any sequence xn → x.  62 Chapter 5 A self-dual approach to inverse problems and optimal control In this chapter we present a unified approach to inverse problems which involve the identification of non-linearities in a given family of monotone PDE. The approach is based on work by Barbu and Kunisch ([BK96] and [BK95]), distilled into a very general form which is perfectly suited to the existing variational theory developed by Ghoussoub in [Gho08a]. The approach relies on a penalization method, where the penalty term is given by a variational potential naturally corresponding to the equation itself (see section 3.6). This approach is easily generalized to more gen- eral optimal control problems. More specific applications are provided in the next chapter. 5.1 An inverse problem via a variational, penalization based approach To better illustrate the general type of problem and the spirit of the general results, let us outline briefly a simple inverse problem and the approach taken in [BK96]: For a convex function Φ, which we will assume is Frechet differentiable, con- 63 sider the problem −∆u+DΦ(u) = f x ∈Ω. u = 0 x ∈ ∂Ω. (5.1) This problem can be shown to have a solution using classic variational meth- ods. Simply consider the functional J̃(u) :=Φ(u)+ 1 2 ∫ Ω |∇u|2− ∫ Ω f u which can be shown to have a minimizer uΦ in the space H10 (Ω) and that mini- mizer corresponds precisely to a solution of the differential equation (5.1). This is the classic direct problem. Observe here that as an alternative to J̃ we could have considered, in view of Fenchel-Young duality (2.2), the functional J(u) :=Φ(u)+Φ∗(∆u+ f )−〈u,∆u+ f 〉= =Φ(u)+Φ∗(∆u+ f )+ ∫ Ω |∇u|2− ∫ Ω f u. At first contact, this might seem like a needlessly complicated alternative, but J has the following desirable properties, which will be fundamental in the following: 1. J(u)≥ 0. 2. J(u) = 0 if and only if u solves (5.1). Suppose we are given a classK of convex functions, and for eachΦ∈K we have a PDE, given by (5.1). The inverse problem consists of the following: Given some function u0, can we recover a function Φ such that u0 corresponds to the solution of the corresponding differential equation (5.1)? There is an important restriction we must observe first: that it may not be possi- ble to find such aΦ, depending on the chosen u0 (maybe u0 is a noisy measurement of the actual solution, for example). But we can still hope to find the closest solu- tion by some least square criterion, that is, a functionΦ such that its corresponding 64 solution uΦ minimizes the functional g(u) := ∫ Ω |u0−u|2 dx. (5.2) This leads to the following minimization problem: find Φ∗ such that if u∗ is its corresponding solution to (5.1), then ‖u0−u∗‖2 = inf{‖u0−u‖2 : u solves (5.1) for some Φ ∈K }. (5.3) The constraint above makes the minimization problem cumbersome to handle, which is where the choice of variational functional comes into play: Observe first, that the previously defined functional J can be seen as a func- tional on u and Φ variables: J(u,Φ) =Φ(u)+Φ∗(∆u+ f )+ ∫ Ω |∇u|2− ∫ Ω f u. This functional is positive and is directly normalized in such a way that its infimum is actually zero and is achieved only at a solution. We will use this functional to penalize the couple (u,Φ) not being a solution to equation (5.1). Define now Iε(u,Φ) := g(u)+ 1 ε J(u,Φ). The functional Iε can be minimized for each fixed ε to obtain (uε ,Φε), note that J(uε ,Φε) is not necessarily zero, which means the pair (uε ,Φε) may not be a solu- tion to (5.1), however, as ε goes to zero, it is enforced that J(uε ,Φε) goes to zero, which, if a limit exists, would yield a solution. This approach will be formalized in the coming results. 5.2 Main result The general problem and approach will be outlined before stating the result. Suppose we are given L , a family of self-dual Lagrangians. This family cor- responds to a family of monotone equations given by: for some fixed p ∈ X∗, for 65 each L ∈L we have p ∈ ∂L(x). (5.4) Given a fixed x0 ∈ X , we are interested on identifying a particular L∗ in L such that if x∗ satisfies the corresponding equation (5.4), then this solution is the best possible approximation to x0, in the sense that it satisfies ‖x0− x∗‖2 = inf{‖x0− x‖2 : p ∈ ∂L(x) for some L ∈L }. (5.5) To overcome the complicated constraint space given above, we will make use of the following functional, given naturally by the self-dual potential (see section 3.6): J(x,L) := L(x, p)−〈x, p〉. (5.6) This functional, in view of (3.1) and definition 3.2, satisfies 1. J(x,L)≥ 0. 2. J(x,L) = 0 if and only if the pair (x,L) solves (5.4). The idea is to then obtain approximate solutions by minimizing the functional Iε(x,L) := ‖x0− x‖2+ 1ε J(x,L). (5.7) This functional’s important qualities are summarized below. Proposition 5.1 The functional Iε is everywhere non negative, and (x∗,L∗) are such that Iε(x∗,L∗) = 0 if and only if both 1. x∗ = x0 2. p ∈ ∂L∗(x0). Proof: Iε is the sum of ‖x0−x‖2 and 1ε ( L(x, p)−〈x, p〉), both positive functionals, and hence is clearly positive. Also, it is zero if and only if both of the previous are zero, which yields the conclusion.  The above means that if the given x0 is achieved inL , meaning that x0 solves (5.4) for some L ∈L , then the minimum value of Iε , for any ε > 0, is actually zero and vice versa! 66 If on the contrary, x0 is not achievable in the classL then the hope is to obtain a best possible approximate solution as a limit of the minimizers (xε ,Lε) by making ε → 0, which should enforce J(xε ,Lε)→ 0, then if a limit for (xε ,Lε) exists, it would be to a minimizing solution. Indeed, the following holds: Theorem 5.2 Let p ∈ X∗ be given. Assume that the class L is Mosco compact. Assume also that for some L ∈L , problem (5.4) has a solution. Fix x0 ∈ X. Then there exists L∗ ∈L such that a corresponding solution to (5.4), x∗, is the best possible approximation to x0 in the sense that it satisfies (5.5). Furthermore, the pair (x∗,L∗) can be obtained as a limit as ε→ 0 of minimizers of the functional Iε . Proof: The functional Iε is non-negative, thus, we can consider a minimizing se- quence {(xn,Ln)} for Iε . Since J is positive, we have that the term ‖x0− x‖2 must remain bounded, since the sequence is minimizing. Hence we can extract from {xn} a weakly convergent subsequence, denoted by the same symbol and converg- ing to some xε ∈ X . Also because of the compactness of the class L , there exists some subsequence of {Ln} denoted by the same symbol, Mosco converging to some Lε ∈L . Then, from the Mosco convergence, Lε(xε , p) ≤ liminfLn(xn, p). This gives, since the norm is weakly lower semicontinuous and 〈·, p〉 is weakly continuous, that Iε(xε ,Lε)≤ liminf Iε(xn,Ln) = inf L×X Iε(x,L), hence Iε(xε ,Lε) = inf L×X Iε(x,L). Now make ε → 0. Observe that since for some L ∈ L there is some x that satisfies p ∈ ∂L(x), then for any ε > 0: Iε(xε ,Lε)≤ Iε(x,L) = ‖x0− x‖2, so as ε → 0, we get, similar as before, that ‖xε‖ remains bounded. Furthermore, since the bound in the previous expression does not depend on ε we must have that 67 as ε → 0 Jε(xε ,Lε) = Lε(xε , p)−〈xε , p〉 → 0. Again, by Mosco compactness of the class L we get that there exist some x∗ ∈ X and L∗ ∈L such that at least for some sequence {εn} converging to zero: xεn ⇀ x∗ weakly in X and L∗(x∗, p)≤ liminfLεn(xεn , p). Now, since 〈·, p〉 is weakly continuous on X : L∗(x∗, p) = 〈x∗, p〉 which yields both p ∈ ∂L∗(x∗) and for any (x,L) ∈ X×L that solves (5.4): ‖x0− x∗‖2 = Iε(x∗,L∗)≤ Iε(x,L) = ‖x0− x‖2, which corresponds to (5.5). The proof is complete.  Remark 5.3 By proposition 5.1, if x0 is achievable in the class, meaning that there is some L0 ∈ L such that the pair (x0,L0) solves (5.4), then infX×L Iε = 0 = Iε(x0,L0). This means that any minimizer in the approximate problem will be the optimal solution, and there is no need to take the limit ε → 0. 5.3 Main results for optimal control problems The approach taken in the previous section can be generalized. We present now analogous results in the context of Optimal Control. Consider first a fixed, bounded below, lower semicontinuous cost g : X → R. 68 The optimal control problem associated to this cost and the family of equations parametrized by L given by (5.4) is to find an L∗ ∈ L providing the cheapest solution for the cost g. That is, to identify L∗ such that the pair (x∗,L∗) satisfies (5.4) and also satisfies g(x∗) = inf{g(x) : p ∈ ∂L(x) for some L ∈L }. (5.8) As before, the complicated constraint set above will be dealt with by instead minimizing the following functional: Iε(x,L) := g(x)+ 1 ε J(x,L), (5.9) where J is defined again by (5.6). Theorem 5.4 Let p ∈ X∗ be given. Assume g is a bounded below, weakly lower semicontinuous and coercive cost in X. Assume that the class L is Mosco com- pact. Assume also that for some L ∈L , problem (5.4) has a solution, denoted xL. Then there exists L∗ ∈L such that the corresponding solution to (5.4), x∗, satis- fies (5.8). Furthermore, the pair (x∗,L∗) can be obtained as a limit as ε → 0 of minimizers of (5.9). Proof: This proof is very similar to the one for Theorem 5.2. The function g(x) takes the place of ‖x−x0‖2 and has the analogue properties needed in the proof.  The coercivity condition can be transferred from the cost functional to the class of potentials: The coercivity condition on g can be removed if the classL is known to be equicoercive in the space variable. Definition 5.5 A class of functionalsF is said to be equicoercive if for some fixed constants M,C > 0 and a> 1 and every F ∈F , F : X → R, the following holds: ‖x‖>M⇒C(‖x‖a−1)≤ F(x). The following holds: Theorem 5.6 Let p ∈ X∗ be given. Assume g is a bounded below, weakly lower semicontinuous cost in X. Assume that the classL isMosco compact and equico- 69 ercive in the first variable. Assume also that for some L ∈L , problem (5.4) has a solution, denoted xL. Then there exists L∗ ∈L such that the corresponding solution to (5.4), x∗, satis- fies (5.8). Furthermore, the pair (x∗,L∗) can be obtained as a limit as ε → 0 of minimizers of (5.9). Proof: As before, since the functional Iε is bounded below, we can consider {(xn,Ln)} a minimizing sequence for Iε . Since {g(xn)} is bounded below, J(xn,Ln)= Ln(xn, p)− 〈xn, p〉 remains bounded. This implies that {xn} is bounded in X , since L is equicoercive. There exist, then, a weakly convergent subsequence denoted again by {xn} converging to some xε ∈ X . The rest of the proof re- mains intact: Since the class L is assumed Mosco compact, there exists some Lε such that for some subsequence of {Ln}, denoted by the same symbol, one has Lε(xε , p)≤ liminfLn(xn, p). This, along with the weak lower semicontinuity of g, gives Iε(xε ,Lε)≤ liminf Iε(xn,Ln) and so Iε(xε ,Lε) = inf X×L Iε(x,L). Now let ε → 0. Since for some L ∈ L there is some x satisfying L(x, p)− 〈x, p〉= 0, we have Iε(xε ,Lε)≤ Iε(x,L) = g(x) which gives a uniform bound, implying that {xε} is bounded in X as ε→ 0. It also implies that Lε(xε , p)−〈xε , p〉 → 0. As before, we can thus assume the existence of (x∗,L∗) ∈ X ×L such that, at least for some sequence {εn} converging to zero, xεn ⇀ x∗ weakly in X and L0(x∗, p) ≤ liminfLε(xε , p). Thus, we have that p ∈ ∂L∗(x∗), and for any pair (x,L) ∈ X×L satisfying (5.4): g(x∗)≤ liminf Iε(xε ,Lε)≤ Iε(x,L) = g(x), which completes the proof.  70 5.4 Further generalizations and comments on coercivity and compactness conditions Also, instead of fixing p ∈ X∗, we can also associate a cost functional to it: h : X∗→ R. The problem consists now on finding a triplet (x∗, p∗,L∗) solving (5.4) such that g(x∗)+h(p∗) = inf{g(x)+h(p) : p ∈ ∂L(x) for some L ∈L }. (5.10) However, there is an issue that must be observed; it may be apparent that asking g and h to be coercive along with the compactness ofL is enough to warranty exis- tence of such an optimal solution, but if the proof is attempted one gets sequences {xn} and {pn} which are only weakly convergent. The problem now is that the limit behaviour of 〈xn, pn〉 cannot be controlled. What is happening, in terms of the Mosco topology, is a loss of compactness in the following way: Allowing p to vary accounts for actually reparametrizing the self-dual potential. Consider the following class M := {M : M(x,q) = L(x,q+ p)−〈x, p〉 for p ∈ X∗ and some L ∈L }, this class contains only self-dual Lagrangians, and solving p ∈ ∂L(x) for some L ∈L , is equivalent to solving 0 ∈ ∂M(x) for some M ∈M . This way the cost on X∗, h(p) can be seen as a cost onM , H(M). However, the classM is not Mosco compact! (The reason being precisely that 〈·, ·〉 cannot be controlled for jointly weakly convergent pairs). To get around this, two approaches will be outlined here: The first is to enforce compactness of the classM by restricting the variable p to some compact class. 71 The second is to ask for stronger compactness conditions on X , via a compactly embedded subspace and keeping a coercivity condition on h. The first alternative is contained in the following: Theorem 5.7 Assume g is a bounded below, weakly lower semicontinuous and coercive cost on X. Let P⊂ X∗ be a compact set. Let h be a bounded below, lower semicontinuous cost on P. Assume that the class L is Mosco compact. Assume also that for some L ∈L and some p ∈ P, problem (5.4) has a solution x ∈ X. Then there exist (x∗, p∗,L∗)∈ X×P×L such that they solve (5.4), and they satisfy (5.10). Furthermore, the triple (x∗, p∗,L∗) can be obtained as a limit as ε → 0 of minimizers of Iε(x,L) := g(x)+h(p)+ 1 ε ( L(x, p)−〈x, p〉). (5.11) Proof: The functional Iε is bounded below, since g and h were assumed bounded below and L(x, p)−〈x, p〉 is a positive functional. Thus we can consider {(xn, pn,Ln)} a minimizing sequence for Iε . Again, since L(x, p)−〈x, p〉 ≥ 0 and h is bounded below, we must have that {g(xn)} is bounded, and by the coercivity of g, this implies that {xn} is bounded in X . There exist, then, a weakly convergent subse- quence, denoted again by {xn}, converging to some xε ∈ X . Since P is compact, we can extract some further subsequence, denoted by the same symbol, such that pn → pε strongly in X∗ for some pε ∈ X∗. Since the class L is assumed Mosco compact, there exists some Lε such that for some subsequence of {Ln}, denoted by the same symbol, one has Lε(xε , pε)≤ liminfLn(xn, pn). This, along with the weak lower semicontinuity of g and h, gives Iε(xε ,Lε) ≤ liminf Iε(xn,Ln) (here we are using that 〈xε , pε〉= lim〈xn, pn〉, which is where the strong convergence of {pn} is required) and so Iε(xε , pε ,Lε) = inf X×X∗×L Iε(x, p,L). Now let ε→ 0. Since for some L∈L there is some x and some p∈P satisfying L(x, p)−〈x, p〉= 0, we have Iε(xε , pε ,Lε)≤ Iε(x, p,L) = g(x)+h(p) which gives a uniform bound, implying that {xε} is bounded in X as ε→ 0. It also implies that Lε(xε , pε)−〈xε , pε〉 → 0. 72 As before, we can thus assume the existence of (x∗, p∗,L∗) ∈ X×X∗×L such that, at least for some sequence {εn} converging to zero, xεn ⇀ x∗ weakly in X , pn → p∗ strongly in X∗ and L∗(x∗, p∗) ≤ liminfLε(xε , pε). Thus, we have that p∗ ∈ ∂L∗(x∗), and for any triple (x, p,L) ∈ X×L satisfying (5.4): g(x∗)+h(p∗)≤ liminf Iε(xε ,Lε)≤ Iε(x,L) = g(x)+h(p), which completes the proof.  The second alternative, to impose growth conditions on a compactly embedded subspace, follows. As in the case of Theorems 5.4 and 5.6, the coercivity condi- tions can be imposed on the cost g or on the classL . Theorem 5.8 Assume Y is a compactly embedded closed subspace of X. Assume that either: g is coercive as a functional on Y and bounded below and lower semi- continuous in X, orL is equicoercive in Y and g bounded below and lower semi- continuous in X. Let h be a bounded below, weakly lower semicontinuous cost on X∗. Assume that the classL isMosco compact. Assume also that for some L ∈L and some p ∈ X∗, problem (5.4) has a solution x ∈ X. Then there exist (x∗, p∗,L∗) ∈ X×X∗×L such that they solve (5.4), and they sat- isfy (5.10). Furthermore, the triple (x∗, p∗,L∗) can be obtained as a limit as ε → 0 of minimizers of (5.11). Proof: Iε is bounded below. Consider then a minimizing subsequence {(xn, pn,Ln)}. By the coercivity condition, {xn} is bounded on Y , which gives the existence of some xε ∈ X such that xn → xε strongly on X . By the coercivity of h, we can recover some pε ∈ X∗ such that (for some subsequence) pn ⇀ pε weakly in X∗. By the Mosco compactness, there exists some Lε ∈ L such that Lε(xε , pε) ≤ liminfLn(xn, pn) and since g and h are lower semicontinuous, we get Iε(xε , pε ,Lε)≤ liminf Iε(xn, pn,Ln) and then Iε(xε , pε ,Lε) = inf L×X Iε(x, p,L). 73 Now let ε → 0. For some (x, p,L) which solves (5.4) this gives Iε(Lε ,xε)≤ Iε(x, p,L) = g(x)+h(p) which implies that, as ε → 0, Lε(xε , pε)−〈xε , pε〉 → 0 and g(xε) bounded, which in turn implies that {xε} is a bounded set in Y , which gives the existence of some x∗ such that, at least for some sequence {εn} converg- ing to zero, xεn → x∗ strongly on X . As before, we can recover some p∗ ∈ X∗ such that for some subsequence pεn ⇀ p∗ weakly in X∗.By using now the Mosco compactness, this gives the existence of some L∗ such that L∗(x∗, p∗)≤ liminfLε(xε , pε) which implies L∗(x∗, p∗)−〈x∗, p∗〉= 0 and by the lower semicontinuity of g and h, we get, for any triple (x, p,L) solving (5.4), g(x∗)+h(p∗)≤ liminf Iε(xε , pε ,Lε)≤ g(x)+h(p), which completes the proof.  74 Chapter 6 Inverse and optimal control problems In this chapter we provide sample applications of the results of the previous chapter. These applications are by no means exhaustive, and are meant to illustrate and explore briefly the generality and usability of the abstract setting developed so far. 6.1 Identification of a convex potential and a transport term Here we consider a slight modification of one of the problems treated in [BK96]. Consider the following (non variational) equation:{ −∆u+b(u) 3 f +~a ·∇u in Ω u = 0 on ∂Ω, (6.1) where ~a :Ω→ Rn is a vector field in Ω with positive divergence and b : R⇒ R is a monotone (possibly multivalued) function. The equation will be parametrized by ~a and b, and the inverse problem will consist of recovering ~a and b from a given solution. We will now put the problem in the context of Theorem 5.2. b will vary in a classB such that for every b ∈B, for some positive constants α1 and α2: α1(|x|−1)≤ b(x)≤ α2(|x|+1), so we will choose to parametrize it by 75 a class of convex function with quadratic bounds by considering ϕ(x) := ∫ x 0 b(t)dt, the function ϕ :R→R is a convex function such that b= ∂ϕ . This allows to define the function Φ : L2(Ω)→ R by Φ(u) := ∫ Ω ϕ ( u(x) ) dx. It is known that Φ as defined above is a convex functional, and its subdifferential map is given by ∂Φ(u) = {g(x) ∈ L2(Ω) : g(x) ∈ ∂ϕ(u(x))= b(u(x)) for a.e. x ∈Ω} (see for example [Gho08a, Proposition 2.7]). Taking the above into consideration, equation (6.1) can be equivalently written as{ −∆u+∂Φ(u) 3 f +~a ·∇u in Ω u = 0 on ∂Ω HenceB will be chosen asB := {∂ϕ(x) : ϕ ∈ F̃}, where F̃ = {ϕ : R→ R | β1|x|2− γ1 < ϕ(x)< β2|x|2+ γ2 and ϕ is convex} and so we can substitute forB, the following class of convex functionals: F = {Φ(u) = ∫ Ω ϕ ( u(x) ) dx : ϕ ∈ F̃}. The field ~a will be chosen among an unspecified class A of positive diver- gence vector fields in Ω which is assumed to be compact with respect to the C∞ topology (for example each component of the vector field smooth on Ω̄ and with all derivatives uniformly bounded in theC norm). The choice for our main Banach space will be L2(Ω), which means all conju- gates will be under the L2 pairing and norms will denote the L2 norm. Also, Mosco convergence will be with respect to the L2 topology. 76 The self-dual Lagrangian associated to equation (6.1) is given by LΦ,~a(u, f ) :=ΨΦ,~a ( u ) +Ψ∗Φ,~a ( f +~a ·∇u+ 1 2 div(~a)u ) , where ΨΦ,~a ( u ) := { +∞, u /∈ H10 (Ω), 1 2‖∇u‖2+Φ(u)+ 14 ∫ Ω div(~a)u 2(x)dx, u ∈ H10 (Ω). (In order to be convinced of the self-duality, observe that ~a ·∇u+ 12div(~a)u is the skew part of the linear operator given by ~a ·∇u, and compare LΦ,~a with item 2 in example 3.8). Remark 6.1 We could have chosen instead of LΦ,~a as defined above, the functional Φ(u)+Φ∗(∆u+~a ·∇u+ f ), which would more closely relate to the original choice made in [BK95, BK96]. This functional is not self-dual, but it is an appropriate representative for equation 6.1 (remember definition 2.19) and it has the needed coercivity conditions. The self-duality condition is not essential in the results from chapter 5. From the above, equation (6.1) is equivalent to f ∈ ∂LΦ,~a(u). With this, we can parametrize the family of equations given by (6.1) by the class of self-dual LagrangiansL , defined by L := {LΦ,~a : Φ ∈F and~a ∈A }. The corresponding inverse problem in the above setting is the following: Given u0 ∈ L2(Ω), find a pair (Φ∗,~a∗)∈F ×A such that for some u∗ the triple (u∗,Φ∗,~a∗) solves (6.1) and it satisfies ‖u0−u∗‖2 = inf{‖u0−u‖2 : u solves (6.1) for some (Φ,~a) ∈F ×A }. The above amounts to finding the optimal L∗ := LΦ∗,~a∗ in the classL . 77 In order to apply Theorem 5.2 to the corresponding inverse problem, we will need to show the following Proposition 6.2 The classL is Mosco compact. The above follows from the Mosco compactness of the class F , the compactness of A and the following two lemmas. Lemma 6.3 Let {Φn} ⊂F and {~an} ⊂A be sequences such that 1. Φn M−→Φ∗. 2. ~an →~a∗ in the C∞ topology. Then ΨΦn,~an M−→ΨΦ∗,~a∗ . Proof: Let us denote Ψn :=ΨΦn,~an and Ψ∗ :=ΨΦ∗,~a∗ . Fix any u ∈ L2(Ω). Since Φn ∈F , lemma 4.30 yields that Φn(u)→ Φ∗(u). Looking at the definition of Ψn and Ψ∗ this trivially gives that Ψn(u)→Ψ∗(u). Let us then move on to prove the liminf property for {Ψn}. Consider a weakly convergent sequence {un}. If it is impossible to extract a subsequence which is bounded in H10 (Ω), then we trivially obtain liminfnΨn(un) = +∞≥Ψ∗(u∗). Let us then assume that the sequence {un} is bounded in H10 (Ω), hence, up to a subsequence, denoted by the same symbol, un→ u∗ strongly in L2(Ω) and un ⇀ u∗ weakly in H10 (Ω), hence, by weak lower semicontinuity of the norm, the strong L2 convergence and the Mosco convergence of {Φn}: liminf n Ψn(un)≤ 12‖∇u∗‖ 2+Φ∗(u∗)+ 1 4 ∫ Ω div(~a)u2∗(x)dx=Ψ∗(u∗). We have established that ΨΦn,~an M−→ΨΦ∗,~a∗ .  Lemma 6.4 Let {Φn} ⊂F and {~an} ⊂A be sequences such that 1. Φn M−→Φ∗. 2. ~an →~a∗ in the C∞ topology. Then LΦn,~an M−→ LΦ∗,~a∗ . 78 Proof: We have established in the previous lemma that under the above hypothe- ses ΨΦn,~an M−→ ΨΦ∗,~a∗ . An analogous argument as that of lemma 4.30, given the growth conditions, will show that for any sequence gn→ g in L2(Ω),Ψ∗Φn,~an(gn)→ Ψ∗Φ∗,~a∗(g). Fix any u inH 1 0 (Ω) and f in L2(Ω), denote gn := f +~an ·∇u+ 12div(~an)u and g := f +~a∗ ·∇u+ 12div(~a∗)u; clearly gn → g in L2(Ω). Hence LΦn,~an(u, f ) =ΨΦn,~an(u)+Ψ ∗ Φn,~an(gn)→ΨΦ∗,~a∗(u)+Ψ∗Φ∗,~a∗(g). The above is LΦn,~an(u, f )→ LΦ∗,~a∗(u, f ), which, combined with self-duality, yields Mosco convergence.  Since given the conditions onF , equation (6.1) always has a solution, we have all the needed conditions to apply Theorem 5.2, from which we have the following Corollary 6.5 There exists Φ∗ ∈F and a∗ ∈A such that for b∗ = ∂Φ, and some u∗ we have both { −∆u∗+b∗(u∗) 3 f +~a ·∇u in Ω u∗ = 0 on ∂Ω, and ‖u0−u∗‖2 = inf{‖u0−u‖2 : u solves (6.1) for some (Φ,~a) ∈F ×A }. Furthermore, the triple (~a∗,Φ∗,u∗) can be recovered as a limit as ε → 0 of the minimizers (~aε ,Φε ,uε) for the functionals Iε(~a,Φ,u) := ‖u0−u‖2+ 1ε ( LΦ,~a(u, f )− ∫ Ω u f dx ) . 79 6.2 Identification of a non conservative diffusion coefficient Consider now the following equation{ −div(T (∇u)) 3 f in Ω u = 0 on ∂Ω (6.2) where the (possibly multivalued) operator T : Rn⇒ Rn is a (maximal) monotone vector field, not necessarily conservative (hence not the gradient of a corresponding potential function). This problem was considered in [BK96] in the case where T is assumed to be the subdifferential of a convex potential. The monotone non linearity T will be assumed to satisfy growth conditions (3.8), that is, (x, p) ∈ T ⇒max{C(‖x‖ 2 2 −1), D(‖p‖ 2 2 −1)} ≤ 〈x, p〉. The corresponding condition for its self-dual Lagrangian LT , constructed as in def- inition 3.2, is (3.9): M(‖x‖2+‖p‖2−1)≤ LT (x, p)≤ N(‖x‖2+‖p‖2+1). As such we will parametrize equation (6.2) with the class T := {T : Rn⇒ Rn : T is maximal monotone and satisfies (3.8)}. We have the corresponding class of self-dual Lagrangians: L̂ := {LT : T ∈T }, and each L ∈ L̂ satisfies growth conditions (3.9). In order to write the self-dual Lagrangian corresponding to equation (6.2), we look at proposition 3.22, and see that FT (u, f ) := inf{ ∫ Ω LT ( ∇u(x),w(x) ) dx;w ∈ L2(Ω;RN),−div(w) = f} 80 is self-dual, and that any solution u of f ∈ ∂FT (u) will be a solution to (6.2). Then, in order to use our results we will consider the class L := {F(u, f ) = inf{ ∫ Ω L ( ∇u(x),w(x) ) dx;w ∈ L2(Ω;RN),−div(w) = f} : L ∈ L̂ }, and show Proposition 6.6 The classL is Mosco compact. Proof: Combining propositions 4.29 and 4.30, given the growth conditions on L̂ , it follows that for any sequence {Ln} we can extract a subsequence such that for some L and for any w ∈ L2(Ω;RN), with −div(w) = f : limsup n Fn(u, f )≤ limsup n ∫ Ω Ln ( ∇u(x),w(x) ) dx= ∫ Ω L ( ∇u(x),w(x) ) dx. Since the above holds for any such w, we get limsup n Fn(u, f )≤ F(u, f ), and we are done.  Then we can simply use Theorem 5.2 to get the existence of an optimal solution to the inverse problem, that is Corollary 6.7 For any given u0 ∈ L2(Ω), there exists T∗ ∈ T such that for some u∗ we have both { −div(T∗(∇u∗)) 3 f in Ω u∗ = 0 on ∂Ω and ‖u0−u∗‖2 = inf{‖u0−u‖2 : u solves (6.2) for some T ∈T }. Furthermore, the couple (T∗,u∗) can be recovered as a limit as ε → 0 of the minimizers (Tε ,uε) of the functional Iε(T,u) := ‖u0−u‖2+ 1ε ( FT (u, f )− ∫ Ω u f dx ) . 81 6.3 Obtaining the cheapest temperature source control for a desired temperature profile Parabolic equations can be handled via self-dual Lagrangians (see [Gho08a] and [Gho08b] for details on how different time-boundary conditions are handled). Now we present the following control problem, involving the heat equation. Consider the heat equation u̇(x, t)−∆u(x, t) = f (x) in Ω× [0,1] u(x, t) = 0 on ∂Ω× [0,1] u(x,0) = g(x) in Ω (6.3) Suppose we are given a specified u0(x, t), with u0(x,0) = g(x), which repre- sents a desired temperature profile to be achieved over Ω along the time interval [0,1]. We can control the temperature by specifying the heat source f over the domain Ω. Assume the cost of maintaining such temperature is given by C( f ) = ∫ 1 0 ‖ f‖2 dt. Assuming we want to achieve an equilibrium between the cost of f and achieve the closest possible behaviour to the profile u0, we want to minimize∫ 1 0 ‖u(x, t)−u0(x, t)‖2+‖ f‖2 dt among all possible solutions u of (6.3) for some f . The self-dual Lagrangian (translated by f ) associated to (6.3) is L f (u,0)= 1 2 ∫ 1 0 ∫ Ω ( |∇u(t,x)|2+ ∣∣∣∇(−∆)−1( f (x)− u̇(t,x))∣∣∣2−2 f (x)u(t,x))dxdt + ∫ Ω |g(x)|2 dx−2 ∫ Ω u(0,x)g(x)dx+ 1 2 ∫ Ω (|u(0,x)|2+ |u(1,x)|2)dx. Equation (6.3), then, is solved by the minimizer of J(u, f ) = L f (u,0). 82 Wemust observe that J(u, f ) is not a self-dual function on (u, f ), see the discussion in section 5.4. We will then assume that f is further restricted to a compact class P⊂ L2(Ω). Fix fi ∈ L2(Ω), i= 1, . . . ,m, and choose P := { f = m ∑ i=1 αi fi : αi ∈ [0,1]}. This class represents the ability to control the heat source by choosing a finite set of parameters. It is obviously compact in L2(Ω). Under this compactness assumption, Theorem 5.7 immediately yields Corollary 6.8 There exists f∗ ∈ P and u∗ such that the pair solves (6.3) and∫ 1 0 ‖u∗(x, t)−u0(x, t)‖2+‖ f∗‖2 dt = inf{ ∫ 1 0 ‖u(x, t)−u0(x, t)‖2+‖ f‖2 dt : (u, f ) solve (6.3) for some f ∈ P}. Furthermore, the couple ( f∗,u∗) can be recovered as a limit as ε → 0 of the mini- mizers ( fε ,uε) of the functional Iε( f ,u) := ‖u0−u‖2+ 1ε J(u, f ). 6.4 Identification of a convex potential and a transport term in a dynamical setting Here we consider the dynamical version of (6.1), that is u̇−∆u+b(u) 3 f +~a ·∇u in Ω× [0,T ] u(x, t) = 0 on ∂Ω× [0,T ], u(x,0) = g(x) in Ω. (6.4) The procedure of moving from stationary to dynamic equations is detailed in [Gho08b, chapter 6], which we follow here. Consider the stationary counterpart of (6.4), namely (6.1). Starting with the self-dual Lagrangian for (6.1), LΦ,~a, we extend this for u in the space A2H10 (Ω) [0,T ] 83 and p = (p1, p0) in the dual of A2H10 (Ω) [0,T ] (identified as H10 (Ω)×L2L2(Ω)) as fol- lows FΦ,~a(u, p) = ∫ 1 0 LΦ,~a ( u(t)+ p0(t),−u̇(t) ) dt+ + 1 4 ‖u(1)−u(0)+ p1‖2+ 〈u(1)−u(0)+ p1,g〉+‖u(0)+u(1)2 +g‖ 2, where the term on the second line is the self-dual Lagrangian corresponding to the boundary conditions being imposed. Choosing p f := ( (−∆−1) f ,0), equation (6.4) is equivalent to equation p f ∈ ∂FΦ,~a(u). As in section 6.1, we will consider ~a ∈ A and b = ∂Φ, with Φ ∈ F . The classes F and A are taken as in section 6.1. In this case the class of self-dual Lagrangians parametrizing the equations is then given by L := {FΦ,~a(u, p) = ∫ 1 0 LΦ,~a ( u(t)+ p0(t),−u̇(t) ) dt+ +S(u(1)−u(0)+ p1, u(0)+u(1)2 ) : Φ ∈F and~a ∈A }. Proposition 6.9 L is Mosco compact. Proof: In the proof of proposition 6.2, the following was established: If Φn M−→Φ∗ and ~an C∞−→ ~a∗, then LΦn,~an(ũ, p̃)→ LΦ∗,~a∗(ũ, p̃). Hence, for such {Φn} and {~an}, for a.e. t ∈ [0,1]: LΦn,~an ( u(t), p(t) )→ LΦ∗,~a∗(u(t), p(t)). Given the boundedness conditions, the dominated convergence theorem now yields ∫ 1 0 LΦn,~an ( u(t), p(t) ) dt → ∫ 1 0 LΦ∗,~a∗ ( u(t), p(t) ) dt. Since F and A are compact in their respective topologies, and given the self- 84 duality of the above functionals, this completes the proof.  We can thus apply Theorem 5.2, which yields Corollary 6.10 Given any u0 ∈ L2L2(Ω), there exists Φ∗ ∈F and a∗ ∈A such that for b∗ = ∂Φ, and some u∗ we have both u̇∗−∆u∗+b(u∗) 3 f +~a ·∇u∗ in Ω× [0,T ] u∗(x, t) = 0 on ∂Ω× [0,T ], u∗(x,0) = g(x) in Ω. and ∫ 1 0 ‖u0(t)−u∗(t)‖2 dt = inf{ ∫ 1 0 ‖u0(t)−u(t)‖2 dt : u solves (6.4) for some (Φ,~a) ∈F ×A }. Furthermore, the triple (~a∗,Φ∗,u∗) can be recovered as a limit as ε → 0 of the minimizers (~aε ,Φε ,uε) of the functional Iε(~a,Φ,u) := ∫ 1 0 ‖u0−u‖2 dt+ 1ε ( FΦ,~a(u, p f )− ∫ 1 0 ∫ Ω u f dxdt ) . 6.5 Identification of a non conservative diffusion coefficient in a dynamical setting Here we consider the dynamical version of (6.2): u̇−div ( T ( ∇xu(x, t) )) 3 f in Ω× [0,1], u(x, t) = 0 on ∂Ω× [0,1], u(x,0) = 0 in Ω. (6.5) What follows is similar to what was done in the previous section, in particular the choice of spaces will be the same. Given the corresponding self-dual Lagrangian 85 for equation (6.2), FT , we lift this to GT (u, p) = ∫ 1 0 FT ( u(t)+ p0(t),−u̇(t) ) dt+ + 1 4 ‖u(1)−u(0)+ p1‖2+ 〈u(1)−u(0)+ p1,g〉+‖u(0)+u(1)2 +g‖ 2, which is a self-dual Lagrangian, corresponding to equation (6.5). As in the previ- ous section, choosing p f := ( (−∆−1) f ,0), equation (6.5) is equivalent to equation p f ∈ ∂GT (u). Then we takeL as in Section 6.2 and consider show that the class G := {GT : FT ∈L }. Proposition 6.11 G is Mosco compact. Proof: In the proof of proposition 6.6 it was established that for any sequence {Fn} in L Mosco converging to F∗, we have: limsupnFn(u, f )≤ F∗(u, f ). This is equivalent to, given the Mosco convergence, lim n Fn(u, f ) = F∗(u, f ). Hence, for such {Fn}, for a.e. t ∈ [0,1]: Fn ( u(t), p(t) )→ F∗(u(t), p(t)). Given the boundedness conditions, the dominated convergence theorem now yields ∫ 1 0 Fn ( u(t), p(t) ) dt → ∫ 1 0 F∗ ( u(t), p(t) ) dt. Since L is also Mosco compact, and given the self-duality of the above function- als, this completes the proof.  We can, again, apply Theorem 5.2, which yields 86 Corollary 6.12 Given any u0 ∈ L2L2(Ω), there exists T∗ ∈ T such that for some u∗ we have both  u̇∗−div ( T ( ∇xu∗(x, t) )) 3 f in Ω× [0,1] u∗(x, t) = 0 on ∂Ω× [0,1] u∗(x,0) = 0 in Ω and∫ 1 0 ‖u0(t)−u∗(t)‖2 dt = inf{ ∫ 1 0 ‖u0(t)−u(t)‖2 dt : u solves (6.5) for some T ∈T }. Furthermore, the couple (T∗,u∗) can be recovered as a limit as ε → 0 of the mini- mizers (Tε ,uε) of the functional Iε(T,u) := ∫ 1 0 ‖u0−u‖2 dt+ 1ε ( GT (u, p f )− ∫ 1 0 ∫ Ω u f dxdt ) . 87 Chapter 7 Homogenization of periodic maximal monotone operators via self-dual calculus In this chapter we consider the homogenization of the problem τn(x) ∈ T ( xεn ,∇un(x)) x ∈Ω, −div(τn(x)) = u∗n(x) x ∈Ω, un(x) = 0 x ∈ ∂Ω, (7.1) where Ω is a bounded domain of RN and T : Ω×RN → RN is a measurable map on Ω×RN such that T (x, ·) is maximal monotone on RN for almost all x ∈ Ω, and such that T (.,ξ ) is Q-periodic for an open non-degenerate parallelogram Q in Rn. This problem has been investigated in recent years by many authors. We refer the interested reader to [Att84, BC94, BCD92, PDD90, DMV07, FMT09, FM87, PD90] for related results. The particular case where the maximal monotone operator is a subdifferential of the form T (x,ξ ) = ∂ξψ(x,ξ ), (7.2) with ψ :Ω×RN →R being a convex function in the second variable is particularly appealing and completely understood. Indeed, under appropriate boundedness and 88 coercivity conditions on ψ , say C0(|ξ |p−1)≤ ψ(x,ξ )≤C1(|ξ |p+1) for all (x,ξ ) ∈Ω×RN , where 1 < p < ∞ and C0,C1 are positive constants, one can use a variational ap- proach to identify for a given u∗ ∈W−1,p(Ω), the solution (u,τ) of (7.1) as the respective minima of the problems inf {∫ Ω ψ(x,∇u(x))dx− ∫ Ω u∗(x)u(x)dx; u ∈W 1,p0 (Ω) } , (7.3) and inf {∫ Ω ψ∗(x,τ(x))dx; div(τ) = u∗ } , (7.4) where ψ∗ is the Fenchel-Legendre dual (in the second variable) of ψ . In this case, the classical concept of Γ-convergence –introduced by DeGiorgi– can be used to show that if u∗n → u∗ strongly inW−1,q(Ω) with q= pp−1 , then up to a subsequence un → u weakly inW 1,p0 (Ω) and τn → τ weakly in Lq(Ω;RN), where u is a solution and τ is a momentum of the homogenized problem{ τ(x) ∈ Thom(∇u(x)) a.e. x ∈Ω, −div(τ(x)) = u∗(x) a.e. x ∈Ω. (7.5) Here Thom can be defined variationally as follows: for ξ ∈RN , Thom(ξ )= ∂ψhom(ξ ), where ψhom(ξ ) := min ϕ∈W 1,p# (Q) 1 |Q| ∫ Q ψ ( x,ξ +∇ϕ(x) ) dx, (7.6) and W 1,p# (Q) = {u ∈W 1,p(Q); ∫ Q u(x)dx= 0 and u is Q−periodic}. (7.7) A similar result can be obtained for general maximal monotone maps T : Ω×RN → RN with appropriate boundedness conditions (see below), by using the more cumbersome graph convergence (or G-convergence, a notion based on clas- sic Kuratowski-Painlevé convergence) methods. In this case, Thom is defined by the 89 following non-variational formula Thom(ξ ) = {∫ Q g(y)dy ∈ RN ;g ∈ Lq#(Q;RN), g(y) ∈ T (y,ξ +∇ψ(y)) a.e. in Q for some ψ ∈W 1,p# (Q) } , (7.8) where Lq#(Q;R N) := { g ∈ Lq(Q;RN); ∫ Q 〈g(y),∇ϕ(y)〉RN dy= 0 for every ϕ ∈W 1,p# (Q) } . (7.9) The goal here is to describe how the variational approach to maximal monotone operators is particularly well suited to deal with the homogenization of such equa- tions, first by showing that –just as in the case of a convex potential (7.2)– the limiting process can be handled through Γ-convergence of associated self dual La- grangians, and secondly by giving a variational characterization for the limiting vector field (7.8) in the same spirit as in (7.6). The following is the main result for this chapter: Theorem 7.1 Let Ω be a domain in RN, q, p > 1 with 1p + 1 q = 1, and assume u∗n → u∗ strongly in W−1,q(Ω). Let un (resp., τn) be (weak) solutions in W 1,p0 (Ω) (resp., momenta in Lq(Ω;RN)) for the Dirichlet boundary value problems (7.1), where T :Ω×RN → RN belongs to MΩ,p(RN). If T (.,ξ ) is Q-periodic for an open non-degenerate parallelogram Q in Rn then, up to a subsequence un → u weakly in W 1,p0 (Ω), τn → τ weakly in Lq(Ω;RN), where u is a solution and τ is a momentum of the homogenized problem τ(x) ∈ Thom(∇u(x)) a.e. x ∈Ω, −div(τ(x)) = u∗(x) a.e. x ∈Ω, u ∈W 1,p0 (Ω). (7.10) Here Thom = ∂̄Lhom, with Lhom being a self dual Lagrangian on RN ×RN defined 90 by Lhom(a,b) := min ϕ∈W 1,p# (Q) g∈Lq#(Q;RN) 1 |Q| ∫ Q L ( x,a+Dϕ(x),b+g(x) ) dx, (7.11) where for each x ∈Ω, L(x, ·, ·) is a self dual Lagrangian on RN×RN such that T (x, ·) = ∂̄L(x, ·). (7.12) We include a homogenization result via Γ-convergence for general Q-periodic Lagrangians which are not necessarily self dual. This is then applied to obtain the result claimed in Theorem 7.1 above in the case of self dual Lagrangians. 7.1 Auxiliary results We open with results used in some of the following proofs. Lemma 7.2 Assume L : Rn×Rn → R is a convex function such that C0(|a|p + |b|q− 1) ≤ L(a,b) ≤ C1(|a|p+ |b|q+ 1) for all a,b ∈ RN where p,q > 1 are two constants. Suppose Ω is a bounded open domain in RN and τ1 ∈ Lp(Ω;RN) and τ2 ∈ Lq(Ω;RN) are two piecewise constant functions such that τ1(x) = ai, x ∈Ωi, and τ2(x) = bi, x ∈Ωi, where {Ωi}i∈I is a finite polyhedral partitions of Ω, and {ai}i∈I,{bi}i∈I are two sequences ∈ RN . Then min f∈Lq(Ω;RN) div f=0 ∫ Ω L ( τ1,τ2(x)+ f (x) ) dx≥∑ i∈I |Ωi| inf ηi∈Rn L(ai,bi+ηi). ProofWe first prove a stronger result (actually an equality) when the set index I is a singleton. For any constant η ∈ RN we have min f∈Lq(Ω;RN) div f=0 ∫ Ω L ( a,b+ f (x) ) dx≤ ∫ Ω L(a,b+η)dx= |Ω|L(a,b+η), 91 from which we obtain min f∈Lq(Ω;RN) div f=0 ∫ Ω L ( a,b+ f (x) ) dx ≤ inf η∈RN |Ω|L(a,b+η), Let now f̃ be the element in Lq(Ω;RN) with div f̃ = 0 such that∫ Ω L ( a,b+ f̃ (x) ) dx= min f∈Lq(Ω;RN)) div f=0 ∫ Ω L ( a,b+ f (x) ) dx. Using Jensen’s inequality, we obtain inf η∈RN |Ω|L(a,b+η) ≤ |Ω|L(a,b+ 1|Ω| ∫ Ω f̃ (x)dx ) = |Ω|L( 1|Ω| ∫ Ω adx, 1 |Ω| ∫ Ω b+ f̃ (x)dx ) ≤ ∫ Ω L ( a,b+ f̃ (x) ) dx = min f∈Lq(Ω;RN) div f=0 ∫ Ω L ( a,b+ f (x) ) dx. This completes the proof when I is a singleton. Now we prove it for the general case. Note first that, using the above argument on each Ωi, we have inf g∈Lq(Ωi;RN) divg=0 ∫ Ωi L ( ai,bi+g(x) ) dx= inf ηi∈RN |Ωi|L(ai,bi+ηi). (7.13) One also can easily deduce that inf f∈Lq(Ω;RN) div f=0 ∫ Ω L ( τ1(x),τ2(x)+ f (x) ) dx≥∑ i inf fi∈Lq(Ωi;RN) div fi=0 ∫ Ωi L ( ai,bi+ fi(x) ) dx.(7.14) 92 In fact if inf f∈Lq(Ω;RN) div f=0 ∫ ΩL ( τ1(x),τ2(x) + f (x) ) dx = ∫ ΩL ( τ1(x),τ2(x) + f̄ (x) ) dx for some f̄ ∈ Lq(Ω;RN) with div( f̄ ) = 0, then∫ Ω L ( τ1(x),τ2(x)+ f̄ (x) ) dx = ∑ i∈I ∫ Ωi L ( ai,bi+ f̄ (x) ) dx ≥ ∑ i∈I inf fi∈Lq(Ωi;RN) div fi=0 ∫ Ωi L ( ai,bi+ fi(x) ) dx. The proof therefore follows from combining (7.13) and (7.14).  The following three Lemmas are standard and we refer to [Suq82] for the proof. Lemma 7.3 Let r ≥ 1 and f ∈ Lr(Q). Then f can be extended by periodicity to a function (still denoted by f ) belonging to Lrloc(RN).Moreover, if (εk) is a sequence of positive real numbers converging to 0 and gk(x) = g( xεk ), then if 1≤ r < ∞, then fk →M( f ) = 1|Q| ∫ Q f (x)dx weakly in L r loc(RN), and if r = ∞, then fk →M( f ) weak∗ in L∞(RN). Lemma 7.4 Let r > 1 and u ∈W 1,r# (Q), then u can be extended by periodicity to an element of W 1,rloc (R N). Lemma 7.5 Let r> 1 and r′= rr−1 . Let g∈Lr ′ (Q;RN) such that ∫ Q〈g(x),∇v(x)〉dx= 0 for every v ∈W 1,r# (Q). Then g can be extended by periodicity to an element of Lr ′ loc(RN ;RN), still denoted by g, such that div(g) = 0 in D′(RN). 7.2 Variational formula for the homogenized maximal monotone vector field Given a maximal monotone family T in MΩ,p(RN) that is Q-periodic for an open non-degenerate parallelogram Q in Rn, its homogenization Thom is normally given by the non-variational formula (7.8). In this section, we shall give a variational formulation for the vector field Thom in terms of a suitably homogenized self dual 93 Lagrangian Lhom derived from the Ω-dependent self dual Lagrangian associated to T . Theorem 7.6 Assume T ∈ MΩ,p(RN) is Q-periodic and let L be an Ω-dependent self dual Lagrangian such that T (x, .) = ∂̄L(x, .) given by Proposition 3.20. If the operator Thom is given by (7.8), then Thom = ∂̄Lhom where Lhom is the self dual Lagrangian on RN×RN given by Lhom(ξ ,η) = min ϕ∈W 1,p# (Q) g∈Lq#(Q;RN) 1 |Q| ∫ Q L ( x,ξ +∇ϕ(x),η+g(x) ) dx. (7.15) The proof relies on the following propositions. First, we show that the homoge- nized Lagrangian Lhom inherits many of the properties of the original Ω-dependent Lagrangian L such as convexity, boundedness and coercivity. Proposition 7.7 Assume L is an Ω-dependent Lagrangian on Ω×RN×RN satis- fying (3.11) for some p,q > 1. Then Lhom is convex and lower semi continuous, and for every a∗,b∗ ∈ Rn, L∗hom(a ∗,b∗) = inf ϕ∈W 1,q′# (Q) g∈Lp′# (Q;RN) 1 |Q| ∫ Q L∗ ( x,a∗+g(x),b∗+∇ϕ(x) ) dx, (7.16) where 1p + 1 p′ = 1 and 1 q + 1 q′ = 1. Furthermore, C0(|a|p+ |b|q−1)≤ Lhom(a,b)≤C1(1+ |a|p+ |b|q) for all a,b∈Rn. (7.17) The following gives the relation between the subdifferentials of Lhom and of L. Proposition 7.8 For each a,b ∈ Rn, the subdifferential map ∂Lhom(a,b) is given by ∂Lhom(a,b) = 1 |Q| ∫ Q ∂L ( y,a+∇ϕ̃(y),b+ g̃(y) ) dy, where ϕ̃ ∈W 1,p# (Q) and g̃ ∈ Lq#(Q;RN) are such that Lhom(a,b) = 1 |Q| ∫ Q L ( y,a+∇ϕ̃(y),b+ g̃(y) ) dy. 94 We need a few preliminary facts. For each 1< r < ∞, set Er := { f = ∇u ∈ Lr(Q;RN); for some u ∈W 1,r# (Q)} and Er+Rn := { f +η : f ∈ Er, η ∈ Rn}. The Poincaré-Wirtenger inequality which states that for D bounded open and con- vex, there exists K := K(r,D)> 0 such that ‖u− 1|D| ∫ D u‖Lr(D) ≤ K‖∇u‖W 1,r(D) for every u ∈W 1,r(D), implies that Er+Rn is a convex weakly closed subset of Lr(Q;RN). The indicator function of Er+Rn, χEr+Rn( f ) = { 0 f ∈ Er+Rn, +∞ f ∈ Lr(Q;RN)\ (Er+Rn), is therefore convex and lower semi-continuous in Lr(Q;RN). Assuming that r′ is the conjugate of r, i.e., 1r + 1 r′ = 1, define E⊥r′ := { g ∈ Lr′(Q;RN); ∫ Ω 〈 f (x),g(x)〉RN dx= 0 for all f ∈ Er+Rn } . The Fenchel-Legendre dual χ∗Er+Rn of χEr+Rn is then given by, χ∗Er+Rn(g) = sup f∈Lr(Q;RN) {∫ Q 〈 f (x),g(x)〉RN dx−χE+Rn( f ) } = sup f∈Er+Rn ∫ Q 〈 f (x),g(x)〉RN dx= χE⊥r′ (g), for all g ∈ Lr′(Q;RN). Also due to the convexity and lower semi-continuity of χEr+Rn one has χ∗E⊥r′ = χEr+Rn . Similarly one can deduce that, χ∗E⊥r′+Rn ( f ) = χEr( f ) 95 for all f ∈ Lr(Q;RN). Note also that Er is the isometric image ofW 1,r# (Q) by ∇ and E⊥r = Lr#(Q;RN). Proof of Proposition 7.7. We first prove (7.16). Fix (a∗,b∗) ∈ Rn×Rn and write L∗hom(a ∗,b∗) = sup (a,b)∈Rn×Rn {〈a,a∗〉RN + 〈b,b∗〉RN −Lhom(a,b)}= = sup (a,b)∈Rn×Rn (ϕ,g)∈W 1,p# (Q)×Lq#(Q;RN) 1 |Q| ∫ Q [ 〈a,a∗〉RN + 〈b,b∗〉RN −L ( x,a+∇ϕ(x),b+g(x) )] dx= = sup (a,b)∈Rn×Rn ( f ,g)∈Ep×E⊥q 1 |Q| ∫ Q [ 〈a,a∗〉RN + 〈b,b∗〉RN −L ( x,a+ f (x),b+g(x) )] dx. Setting A(x) = a+ f (x), B(x) = b+g(x) and substituting above we have L∗hom(a ∗,b∗) = sup A∈Ep+Rn B∈E⊥q +Rn 1 |Q| ∫ Q [ 〈A,a∗〉RN + 〈B,b∗〉RN −L ( x,A(x),B(x) )] dx = sup A∈Lp(Ω;Rn) B∈Lq(Ω;Rn) { 1 |Q| ∫ Q [ 〈A,a∗〉RN + 〈B,b∗〉RN −L ( x,A(x),B(x) )] dx −χEp+Rn(A)−χE⊥q +Rn(B) } . Now using the fact that the Fenchel dual of a sum is their inf-convolution, we obtain L∗hom(a ∗,b∗) = inf f∈Lq′ (Q;Rn) g∈Lp′ (Q;Rn) { 1 |Q| ∫ Q L∗ ( x,a∗−g(x),b∗− f (x))dx+χE⊥p′ (g)+χEq′ ( f )} = inf f∈Eq′ g∈E⊥p′ 1 |Q| ∫ Q L∗ ( x,a∗−g(x),b∗− f (x))dx = inf ϕ∈W 1,q′# (Q) g∈Lp′# (Q;RN) 1 |Q| ∫ Q L∗ ( x,a∗+g(x),b∗+∇ϕ(x) ) dx. This proves (7.16), which then implies that L∗∗hom = Lhom and therefore Lhom is con- vex and lower semi-continuous. 96 We now prove estimate (7.17). In fact, the upper bound simply follows from Lhom(a,b)≤ 1|Q| ∫ Q L(x,a,b)dx≤C1(|a|p+ |b|q+1). For the lower bound, note first that since C0(|a|p + |b|q− 1) ≤ L(x,a,b) for all a,b ∈ RN , it follows that L∗(x,a,b)≤ C0(p−1) (C0p)p ′ |a|p ′ + C0(q−1) (C0q)q ′ |b|q ′ +C0 for all a,b ∈ RN . One then gets from (7.16) that L∗hom(a,b)≤ 1 |Q| ∫ Q L∗(x,a,b)dx≤ C0(p−1) (C0p)p ′ |a|p ′ + C0(q−1) (C0q)q ′ |b|q ′ +C0 for all a,b ∈ RN , from which we get that Lhom(a,b) = L∗∗hom(a,b)≥C0(|a|p+ |b|q−1) for all a,b ∈ RN .  Proof of Proposition 7.8. Setting A(a,b) := 1|Q| ∫ Q ∂L ( y,a+∇ϕ̃(y),b+ g̃(y) ) dy, we shall first show that A ⊂ ∂Lhom. For that consider (a1,b1) ∈ RN ×RN ,ϕ ∈ W 1,p# (Q) and g ∈ Lq#(Q;RN). From the convexity of L: L ( y,a1+∇ϕ(y),b1+g(y) )≥ L(y,a+∇ϕ̃(y),b+ g̃(y)) + 〈∂1L ( y,a+∇ϕ̃(y),b+ g̃(y) ) ,a1+∇ϕ(y)−a−∇ϕ̃(y)〉RN + + 〈∂2L ( y,a+∇ϕ̃(y),b+ g̃(y) ) ,b1+g(y)−b− g̃(y)〉RN . Averaging the above on Q implies that 1 |Q| ∫ Q L ( y,a1+∇ϕ(y),b1+g(y) ) dy≥Lhom(a,b)+〈A(a,b),(a1−a,b1−b)〉RN×RN , from which we get Lhom(a1,b1)≥ Lhom(a,b)+ 〈A(a,b),(a1−a,b1−b)〉RN×RN . 97 This implies that A⊂ ∂Lhom. To prove the reverse inclusion, let (d,c) be in ∂Lhom(a,b). Since Lhom is convex and lower semi-continuous, we have Lhom(a,b)+L∗hom(d,c) = 〈a,d〉RN + 〈b,c〉RN . It follows from Proposition 7.7 that there exist ϕ ∈W 1,q′# (Q) and g ∈ Lp ′ # (Q;RN) such that L∗hom(a ∗,b∗) = 1 |Q| ∫ Q L∗ ( x,a∗+g(x),b∗+∇ϕ(x) ) dx, and therefore 1 |Q| ∫ Q ( L ( y,a+∇ϕ̃(y),b+ g̃(y) ) +L∗ ( x,d+g(x),c+∇ϕ(x) )) dx= 〈a,d〉RN +〈b,c〉RN . On the other hand, 〈a,d〉RN + 〈b,c〉RN = 1 |Q| ∫ Q 〈a+∇ϕ̃(y),d+g(y)〉RN + 〈b+ g̃(y),c+∇ϕ(y)〉RN dy, which together with the previous equality yield∫ Q [ L ( y,a+∇ϕ̃(y),b+ g̃(y) ) +L∗ ( y,d+g(y),c+∇ϕ(y) ) ]dy =∫ Q [〈a+∇ϕ̃(y),d+g(y)〉RN −〈b+ g̃(y),c+∇ϕ(y)〉RN ]dy. Taking into account that, pointwise, on the previous equation, the integrand above is greater then the integrand below almost everywhere, we obtain L ( y,a+∇ϕ̃(y),b+ g̃(y) ) +L∗ ( y,d+g(y),c+∇ϕ(y) ) = 〈a+∇ϕ̃(y),d+g(y)〉RN −〈b+ g̃(y),c+∇ϕ(y)〉RN for almost all y ∈ Q. This implies that (d+g(y),c+∇ϕ(y)) ∈ ∂L(y,a+∇ϕ̃(y),b+ g̃(y)) a.e. y ∈ Q. 98 Integrating the above over Q implies that (d,c) ∈ 1|Q| ∫ Q ∂L ( y,a+∇ϕ̃(y),b+ g̃(y) ) , which completes the proof.  Proof of Theorem 7.6 Let η ∈ ∂̄Lhom(ξ ) in such a way that Lhom(ξ ,η) = 〈ξ ,η〉RN . From the definition of Lhom, we have Lhom(ξ ,η) = min ϕ∈W 1,p# (Q) g∈Lq#(Q;RN) 1 |Q| ∫ Q L ( x,ξ +∇ϕ(x),η+g(x) ) dx. From the coercivity assumptions on L, it follows that there exist ϕ ∈W 1,p# (Q) and g ∈ Lq#(Q;RN) such that Lhom(p,q) = 1 |Q| ∫ Q L ( x, p+Dϕ(x),q+g(x) ) dx. Hence 0 = Lhom(ξ ,η)−〈ξ ,η〉RN = 1 |Q| ∫ Q L ( x,ξ +∇ϕ(x),η+g(x) ) dx−〈ξ ,η〉RN = 1 |Q| ∫ Q [ L ( x,ξ +∇ϕ(x),η+g(x) )−〈ξ +∇ϕ(x),η+g(x)〉RN]dx, and since the integrand in non-negative we obtain L ( x,ξ +∇ϕ(x),η+g(x) )−〈ξ +∇ϕ(x),η+g(x)〉RN = 0 for a.e. x ∈ Q, from which we have η+g(x) ∈ ∂̄L(x,ξ +∇ϕ(x)) = T (x,ξ +∇ϕ(x)) and finally η = ∫ Q(η + g(x))dx. This implies that ∂̄Lhom ⊂ Thom and the equality follows since ∂̄Lhom is itself a maximal monotone operator.  99 7.3 A variational approach to homogenization We start by studying the homogenization of a class of Lagrangians that is more general than the one introduced in Proposition 3.22. We shall then apply this result to deduce Theorem 7.1 announced in the introduction. 7.3.1 The homogenization of general Lagrangians on W 1,p(Ω)×Lq(Ω;RN) The following homogenization result does not require theΩ-dependent Lagrangian L to be self dual nor that the exponents p and q to be conjugate. Theorem 7.9 Let Ω be a regular bounded domain and Q an open non-degenerate parallelogram in Rn. Let L : Ω×RN ×RN → R be an Ω-dependent Lagrangian such that: (1) For each a,b ∈ RN the function x→ L(x,a,b) is Q-periodic. (2) There exist constants C0,C1 ≥ 0 and exponents p,q > 1 such that for every x ∈Ω, C0(|a|p+ |b|q−1)≤ L(x,a,b)≤C1(|a|p+ |b|q+1). Let {Gε ;ε > 0} be the family of functionals on W 1,p(Ω)×Lq(Ω;RN) defined by Gε(u,τ) := inf f∈Lq(Ω;RN) div f=0 ∫ Ω L ( x ε ,∇u(x),τ(x)+ f (x) ) dx, and set Lhom(a,b) := min ϕ∈W 1,p# (Q) g∈Lq#(Q;RN) 1 |Q| ∫ Q L ( x,a+∇ϕ(x),b+g(x) ) dx. (7.18) Equip Lq(Ω;RN) with the following topology denoted by T , τn → τ for T if and only if τn → τ weakly in Lq(Ω;RN) and div(τn)→ div(τ) strongly in W−1,q(Ω), 100 There exists then a Lagrangian Ghom on W 1,p(Ω)×Lq(Ω;RN) that is a Γ-limit of {Gε ;ε > 0} as ε → 0. Moreover, Ghom is given by the formula Ghom(u,τ) := inf f∈Lq(Ω;RN) div f=0 ∫ Ω Lhom ( ∇u(x),τ(x)+ f (x) ) dx, (7.19) Remark 7.10 Note that when the Lagrangian L is independent of the third vari- able, i.e., L(x,a,b) = ϕ(x,a) for all (x,a,b) ∈Ω×RN×RN , for some function ϕ : Ω×RN → R, this homogenization problem is completely understood. Also, when the Lagrangian L is independent of the second variable then this problem can be dealt using the bi-continuity of the Fenchel dual (see for instance [Att84, DMS08]). The proof for the general Lagrangians consists of two parallel parts corresponding to each of these variables and should be done simultaneously for both. The part regarding the first variable is rather standard and the same argument can be found for instance in [Att84]. The proof of Theorem 7.9 will follow from the following two lemmas. Lemma 7.11 For any (u,τ)∈W 1,p(Ω)×Lq(Ω;RN), there exists a sequence (uε ,τε)∈ W 1,p(Ω)× Lq(Ω;RN) such that uε → u strongly in Lp(Ω), τε → τ strongly in Lq(Ω;RN) and limsup ε→0 Gε(uε ,τε)≤ Ghom(u,τ). Lemma 7.12 Let f ∈ Lq(Ω;RN) with div( f ) = 0. For any (u,τ) ∈ W 1,p(Ω)× Lq(Ω;RN) and any sequence (uε ,τε) such that uε → u strongly in Lp(Ω) and τε → τ with the T -topology in Lq(Ω;RN), we have liminf ε→0 ∫ Ω L( x ε ,∇uε(x),τε(x)+ f (x))dx≥ ∫ Ω Lhom ( ∇u(x),τ(x)+ f (x) ) dx. We first show how Theorem 7.9 follows from the two lemmas above. The limsup property in the definition of Γ-convergence readily follows from Lemma 7.11. For the liminf property we must show that for any (u,τ) ∈W 1,p(Ω)×Lq(Ω;RN) and 101 any sequence {(uε ,τε)} ⊂W 1,p(Ω)×Lq(Ω;RN) such that uε → u strongly in Lp(Ω) and τε → τ in theT − topology, we have that liminf ε→0 Gε(uε ,τε)≥ Ghom(u,τ). By Lemma 7.12 we have liminf ε→0 ∫ Ω L( x ε ,∇uε ,τε + f )dx≥ ∫ Ω Lhom ( ∇u,τ+ f ) dx, for every f ∈ Lq(Ω;RN) with div( f ) = 0. Since inf f∈Lq(Ω;RN) div f=0 liminf ε→0 ∫ Ω L( x ε ,∇uε ,τε+ f )dx= liminf ε→0 inf f∈Lq(Ω;RN) div f=0 ∫ Ω L( x ε ,∇uε ,τε+ f )dx, we obtain that liminfε→0Gε(uε ,τε)≥ Ghom(u,τ), as desired.  Proof of Lemma 7.11. Note that without loss of generality we may assume L≥ 0. Assume first that u is an affine function and τ is constant on Ω, that is u(x) = 〈a,x〉+α and τ(x) = b, for some a and b in Rn and α ∈R. Fix η ∈Rn and let ϕ̃ and g̃ to be the minimizers on the formula for Lhom given by (7.18): Lhom(a,b+η) = 1 |Q| ∫ Q L ( x,a+∇ϕ̃(x),b+η+ g̃(x) ) . (7.20) Define uε(x) := u(x)+ εϕ̃( xε ) and τε(x) := τ. 102 Note that by Lemma 7.5 in section 7.1, g̃ can be extended by periodicity to an element of Lqloc(R N ;RN), still denoted by g̃ such that div(g̃) = 0. It follows that limsup ε Gε(uε ,τε) = limsup ε inf f∈Lq(Ω;RN) div f=0 ∫ Ω L ( x ε ,a+∇ϕ̃( x ε ),b+ f (x) ) dx ≤ inf f∈Lq(Ω;RN) div f=0 limsup ε ∫ Ω L ( x ε ,a+∇ϕ̃( x ε ),b+ f (x) ) dx ≤ limsup ε ∫ Ω L ( x ε ,a+∇ϕ̃( x ε ),b+η+ g̃( x ε ) ) dx. By Lemma 7.3 of section 7.1 we have as ε → 0, ∫ Ω L ( x ε ,a+∇ϕ̃( x ε ),b+η+ g̃( x ε ) ) dx→ |Ω||Q| ∫ Q L ( y,a+∇ϕ̃(y),b+η+ g̃(y) ) dy. It then follows from (7.20) that limsup ε→0 Gε(uε ,τε)≤ |Ω|Lhom(a,b+η), and since η is arbitrary, we have that limsup ε→0 Gε(uε ,τε)≤ inf η∈Rn |Ω|Lhom(a,b+η). By Lemma 7.2 we have inf f∈Lq(Ω;RN) div f=0 ∫ Ω Lhom ( a,b+ f (x) ) dx≥ inf η∈Rn |Ω|Lhom(a,b+η), and thus we conclude, as desired limsup ε→0 Gε(uε ,τε)≤ inf f∈Lq(Ω;RN) div f=0 ∫ Ω Lhom ( a,b+ f (x) ) dx= Ghom(u,τ). Assume now that u is a piecewise affine function and τ is a piecewise constant function on Ω, that is for {Ω̂ j} j∈I1 and {Ω̃k}k∈I2 , both finite polyhedral partitions 103 of Ω, we have u(x) = 〈a j,x〉+α j for x ∈ Ω̂ j and τ(x) = bk for x ∈ Ω̃k, for fixed a j ∈ Rn and bk ∈ Rn and constants α j. By considering non-empty in- tersections Ω̂ j ∩ Ω̃k and re-indexing them, we can consider {Ωi}i∈I a polyhedral partition of Ω such that u(x) = 〈ai,x〉+αi for x ∈Ωi and τ(x) = bi for x ∈Ωi. Analogous to what was done previously, fix {ηi} ⊂ RN and let ϕ̃i and g̃i be such that Lhom(ai,bi+ηi) = 1 |Q| ∫ Q L ( x,ai+∇ϕ̃i(x),bi+ηi+ g̃i(x) ) dx, and set uiε(x) := u(x)+ εϕ̃i( xε ). Unfortunately, we cannot consider uε as the piecewise construction of the above functions, as the ϕi won’t necessarily match along the interface between the Ωi and thus will not in general be a function in W 1,p(Ω). This can be reme- died by the following standard construction (see for instance [Att84]): Let Σ be the interface set between the Ωi, and define for δ > 0, Σδ := {x ∈ Ω : d(x,Σ) ≤ δ}. Consider a smooth function Ψδ so that Ψδ (x) = { 1 x ∈ Σδ 0 x ∈Ω\Σ2δ , and define uδε (x) := ( 1−Ψδ (x) ) uiε(x)+Ψδ (x)u(x) for x ∈Ωi and τε := τ. It can be checked that the function uδε lies in W 1,p(Ω). Note that by Lemma 7.5, each g̃i can be extended by periodicity to an element of L q loc(R N ;RN), still denoted by g̃i such that div(g̃i) = 0. Thus div(ηi+ g̃i( xε )) = 0 on R N and in particular on Ωi\Σδ .Define fε,δ (x)=ηi+ g̃i( xε ) onΩi\Σδ .One can also extend (using Theorem 2.5 and Corollary 2.8 in [VG86]) fε,δ to an element in Lq(Ω;RN), still denoted by fε,δ such that ‖ fε,δ‖Lq(Ω;RN) is bounded and div( fε,δ ) = 0. Take now any 0< t < 1, 104 then Gε(tuδε ,τε) = inf f∈Lq(Ω;RN) div f=0 ∫ Ω L ( ∇tuδε ,τε + f ) dx ≤ ∫ Ω L ( ∇tuδε ,τε + fε,δ ) dx= ∑ i ∫ Ωi\Σδ L ( x ε , t ( 1−Ψδ ) ∇uiε+tΨδ∇u+ t(1− t) (1− t) (u−u i ε)∇Ψδ ,bi+ηi+ g̃i( x ε ) ) dx+ + ∫ Σδ L ( ∇tuδε ,τε + fε,δ ) dx Since L is convex in the middle variable and since t(1−Ψδ )+ tΨδ +(1− t) = 1, we obtain Gε(tuδε ,τε) ≤ ∑ i ∫ Ωi\Σδ t(1−Ψδ )L ( x ε ,ai+∇ϕ̃i( x ε ),bi+ηi+ g̃i( x ε ) ) dx + ∑ i ∫ Ωi\Σδ (1− t)L ( x ε , t (1− t)(u−u i ε)∇Ψδ ,bi+ηi+ g̃i( x ε ) dx + ∫ Σ2δ \Σδ tΨδL ( x ε ,∇u,bi+ηi+ g̃i( x ε ) dx + ∫ Σδ L ( ∇tuδε ,τε + fε,δ ) dx. For the first term on the right hand side of this inequality we have∫ Ωi\Σδ t(1−Ψδ )L ( x ε ,ai+∇ϕ̃i( x ε ),bi+ηi+ g̃i( x ε ) ) dx≤ ∫ Ωi\Σδ L ( x ε ,ai+∇ϕ̃i( x ε ),bi+ηi+ g̃i( x ε ) ) dx. Using the boundedness of L we get the following estimate for the second term,∫ Ωi\Σδ (1− t)L ( x ε , t (1− t)(u−u i ε)∇Ψδ ,bi+ηi+ g̃i( x ε ) ) dx≤ C1(1− t) ∫ Ωi\Σδ ( | t (1− t)(u−u i ε)∇Ψδ |p+ |bi+ηi+ g̃i( x ε )|q+1 ) dx, 105 and similarly∫ Σ2δ \Σδ tΨδL ( x ε ,∇u,bi+ηi+ g̃i( x ε ) ) dx≤C1 ∫ Σ2δ \Σδ ( 1+ |∇u|p+ |bi+ηi+ g̃i( xε )| q ) dx, as well as∫ Σδ L ( ∇tuδε ,τε + fε,δ ) dx≤C1 ∫ Σδ ( 1+ |∇tuδε |p+ |τε + fε,δ |q ) dx. It then follows that Gε(tuδε ,τε) ≤ ∑ i ∫ Ωi\Σδ L ( x ε ,ai+∇ϕ̃i( x ε ),bi+ηi+ g̃i( x ε ) ) dx + C1(1− t)∑ i ∫ Ωi\Σδ ( | t(u−u i ε) (1− t) ∇Ψδ | p+ |bi+ηi+ g̃i( xε )| q+1 ) dx + C1 ∫ Σ2δ \Σδ ( 1+ |∇u|p+ |bi+ηi+ g̃i( xε )| q ) dx + C1 ∫ Σδ ( 1+ |∇tuδε |p+ |τε + fε,δ |q ) dx. By taking limsupε→0 on both sides and considering uiε → u on Lp(Ωi), and then letting t → 1 and δ → 0 we finally get, limsup t→1 δ→0 limsup ε→0 Gε(tuδε ,τε)≤∑ i |Ωi| |Q| ∫ Q L ( x,ai+∇ϕ̃i(x),bi+ηi+ g̃i(x) ) dx. (7.21) Also note that, ∑ i |Ωi| |Q| ∫ Q L ( x,ai+∇ϕ̃i(x),bi+ηi+ g̃i(x) ) dx=∑ i |Ωi|Lhom(ai,bi+ηi). A diagonalization argument yields from limit (7.21) the existence of some t(ε) and δ (ε) such that t(ε)→ 1 and δ (ε)→ 0 as ε → 0. Defining uε := t(ε)uδ (ε)ε , we obtain limsup ε→0 Gε(uε ,τε)≤∑ i |Ωi|Lhom(ai,bi+ηi), 106 and since the {ηi} is arbitrary one has limsup ε→0 Gε(uε ,τε)≤∑ i |Ωi| inf ηi∈Rn Lhom(ai,bi+ηi). Now we use Lemma 7.2 to obtain ∑ i |Ωi| inf ηi∈Rn Lhom(ai,bi+ηi)≤ inf f∈Lq(Ω;RN) div f=0 ∫ Ω Lhom ( ∇u(x),τ(x)+ f (x) ) dx, from which we get limsupε→0Gε(uε ,τε)≤ Ghom(u,τ). Finally, consider any (u,τ) ∈W 1,p(Ω)× Lq(Ω;RN). There exists then a se- quence {un} of piecewise affine functions and a sequence {τn} of piecewise con- stant functions such that (un,τn)→ (u,τ). By Proposition 7.7, the function Ghom are continuous, so we also have lim n Ghom(un,τn) = Ghom(u,τ). For each n, we have shown the existence of (uεn,τεn ) such that uεn → un and τεn → τn in Lp(Ω) and Lq(Ω;RN) respectively and limsup ε→0 Gε(uεn,τ ε n )≤ Ghom(un,τn), so we get limsup n limsup ε→0 Gε(uεn,τ ε n )≤ Ghom(u,τ). From the same diagonalization argument as before, there exists some n(ε) such that n(ε)→ ∞ as ε → 0 for which, by defining (uε ,τε) := (uεn(ε),τεn(ε)) we obtain uε → u strongly in Lp(Ω), τε → τ strongly in Lq(Ω;RN) and limsup ε→0 Gε(uεn,τ ε n )≤ Ghom(u,τ). This concludes the proof of Lemma 7.11.  107 Proof of Lemma 7.12. Let (u,τ) ∈W 1,p(Ω)×Lq(Ω;RN) and f ∈ Lq(Ω;RN) with div( f ) = 0. We assume that uε → u strongly in Lp(Ω) and τε → τε in T . For constant vectors ai,bi,ηi ∈ Rn, consider as before functions ϕ̃i ∈W 1,p# (Q) and g̃i ∈ Lq#(Q;RN) such that Lhom(ai,bi+ηi) = 1 |Q| ∫ Q L ( x,ai+∇ϕ̃i(x),bi+ηi+ g̃i(x) ) dx. Denote ∂1L the subdifferential of L with respect to the middle variable and ∂2L the subdifferential of L with respect to the last variable. From the above we have both div ( ∂1L ( y,ai+Dϕ̃i(y),bi+ηi+ g̃i(y) )) = 0 a.e. y ∈ Q, (7.22) and ∫ Q 〈∂2L ( y,ai+Dϕ̃i(y),bi+ηi+ g̃i(y) ) ,g(y)〉dy= 0, (7.23) for any g ∈ Lq#(Q;RN). It follows from (7.23) that ∂2L ( y,ai+∇ϕ̃i(y),bi+ηi+ g̃i(y) ) = ∇w(y) a.e. y ∈ Q, (7.24) for some w ∈W 1,p# (Q). It also follows from Lemma 7.4 that w can be extended by periodicity to an element inW 1,ploc (R N). Now, let û ∈W 1,p(Ω) be a piecewise affine functions and τ̂ ∈ Lq(Ω;RN) be a piecewise constant function such that for some partition {Ωi} of Ω we have û(x) = 〈ai,x〉+αi for x ∈Ωi and τ̂(x) = bi for x ∈Ωi. Consider now for x ∈Ωi, ûε(x) := û(x)+ εϕ̃i( xε ) and τ̂ε(x) := τ̂(x). 108 From the convexity of L we get L ( x ε ,∇uε(x),τε(x)+ f (x) )≥ L( x ε ,∇ûε(x), τ̂ε(x)+ηi+ g̃i( x ε ) ) +〈∂1L ( x ε ,∇ûε(x), τ̂ε(x)+ηi+ g̃i( x ε ) ) ,∇uε(x)−∇ûε(x)〉 +〈∂2L ( x ε ,∇ûε(x), τ̂ε(x)+ηi+ g̃i( x ε ) ) ,τε(x)− τ̂ε(x)〉 +〈∂2L ( x ε ,∇ûε(x), τ̂ε(x)+ηi+ g̃i( x ε ) ) , f (x)−ηi− g̃i( xε )〉. Consider now smooth functions Ψi : Ωi → R with compact support such that 0<Ψi < 1. Multiplying the above convexity inequality by Ψi, integrating over Ωi and adding over all i, we get the following:∫ Ω L( x ε ,∇uε ,τε + f )dx≥∑ i ∫ Ωi L ( x ε ,ai+∇ϕ̃i( x ε ),bi+ηi+ g̃i( x ε ) ) Ψi(x)dx +∑ i ∫ Ωi 〈∂1L ( x ε ,ai+∇ϕ̃i( x ε ),bi+ηi+ g̃i( x ε ) ) ,∇uε(x)−∇ûε(x)〉Ψi(x)dx +∑ i ∫ Ωi 〈∂2L ( x ε ,ai+∇ϕ̃i( x ε ),bi+ηi+ g̃i( x ε ) ) ,τε(x)− τ̂ε(x)〉Ψi(x)dx +∑ i ∫ Ωi 〈∂2L ( x ε ,ai+∇ϕ̃i( x ε ),bi+ηi+ g̃i( x ε ) ) , f (x)−ηi〉Ψi(x)dx +∑ i ∫ Ωi 〈∂2L ( x ε ,ai+∇ϕ̃i( x ε ),bi+ηi+ g̃i( x ε ) ) ,−g̃i( xε )〉Ψi(x)dx. Now we deal with each term independently. For the first term on the right hand side of the above expression we have∫ Ωi L ( x ε ,ai+∇ϕ̃i( x ε ),bi+ηi+ g̃i( x ε ) ) Ψi(x)dx→ ∫ Ωi Lhom(ai,bi+ηi)Ψi(x)dx, by virtue of Lemma 7.3. For the second term, by integrating by parts and by then taking into account (7.22) we obtain∫ Ωi 〈∂1L ( x ε ,ai+∇ϕ̃i( x ε ),bi+ηi+ g̃i( x ε ) ) ,∇uε(x)−∇ûε(x)〉Ψi(x)dx =− ∫ Ωi 〈∂1L ( x ε ,ai+∇ϕ̃i( x ε ),bi+ηi+ g̃i( x ε ) ) ,(uε − ûε)∇Ψi(x)〉dx. 109 It follows from Lemma 7.3 and Proposition 7.8 below, that if ε → 0 then,∫ Ωi 〈∂1L ( x ε ,ai+∇ϕ̃i( x ε ),bi+ηi+ g̃i( x ε ) ) ,(uε − ûε)∇Ψi(x)〉dx→∫ Ωi 〈∂1Lhom(ai,bi+ηi),(u− û)∇Ψi(x)〉dx Integrate by parts on more time to get∫ Ωi 〈∂1Lhom(ai,bi+ηi),(u− û)∇Ψi(x)〉dx= − ∫ Ωi 〈∂1Lhom(ai,bi+ηi),∇u−∇û〉Ψi(x)dx, from which one has∫ Ωi 〈∂1L ( x ε ,ai+∇ϕ̃i( x ε ),bi+ηi+ g̃i( x ε ) ) ,∇uε(x)−∇ûε(x)〉Ψi(x)dx→∫ Ωi 〈∂1Lhom(ai,bi+ηi),∇u−∇û〉Ψi(x)dx. For the third term, we use (7.24) to get ∂2L ( x ε ,ai +Dϕ̃i( x ε ),bi + ηi + g̃i( x ε ) ) = ∇w( xε ) for some w ∈W 1,p# (Q). Using an integration by parts, we obtain∫ Ωi 〈∇w( x ε ),τε(x)− τ̂ε(x)〉Ψi(x)dx = − ∫ Ωi εw( x ε )div ( τε(x)− τ̂ε(x) )〉Ψi(x)dx − ∫ Ωi εw( x ε )〈∇Ψi(x),τε(x)− τ̂ε(x)〉dx, which goes to 0 as ε → 0 since τε → τ in the T -topology. Similarly as above, the fourth term can be seen to converge to∫ Ωi 〈∂2Lhom(ai,bi+ηi), f (x)−ηi〉Ψi(x)dx, while for the fifth term, we first observe that the function mi(x) := 〈∂2L ( x,ai+∇ϕ̃i(x),bi+ηi+ g̃i(x) ) , g̃i(x)〉 110 is Q-periodic, and thus setting (mi)ε(x) := mi( xε ), it follows from Lemma 7.3 that (mi)ε ⇀ mi weakly in L1, where mi = 1 |Q| ∫ Q 〈∂2L ( y,ai+∇ϕ̃i(y),bi+ηi+ g̃i(y) ) ,−g̃i(y)〉dy, which is equal to 0 in view of (7.23). The fifth term therefore disappears as ε → 0. Putting now all of the above together we obtain that liminf ε→0 ∫ Ω L( x ε ,∇uε ,τε + f )dx ≥∑ i ∫ Ωi Lhom(ai,bi+ηi)Ψi(x)dx +∑ i ∫ Ωi 〈∂1Lhom(ai,bi+ηi),∇u(x)−∇û(x)〉Ψi(x)dx +∑ i ∫ Ωi 〈∂2Lhom(ai,bi+ηi), f (x)−ηi〉Ψi(x)dx. By taking into account the estimate |∂Lhom(a,b)| ≤M(1+ |a|p−1+ |b|q−1) for all a,b ∈ RN , which follows from estimate (7.17) in Proposition 7.7, and letting Ψi ↑ 1 on each Ωi, it follows from the dominated convergence theorem that liminf ε→0 ∫ Ω L( x ε ,∇uε ,τε + f )dx ≥ ∫ Ω Lhom ( ∇û(x), τ̂(x)+ f̃ (x) ) dx + ∫ Ω 〈∂1Lhom ( ∇û(x), τ̂(x)+ f̃ (x) ) ,∇u(x)−∇û(x)〉dx + ∫ Ω 〈∂2Lhom ( ∇û(x), τ̂(x)+ f̃ (x) ) , f (x)− f̃ (x)〉dx. where f̃ ∈ Lq(Ω;RN) is a function defined by f̃ (x) = ηi on Ωi. The above is valid for arbitrary piecewise affine function û, and piecewise constant functions τ̂, f̃ . We can then let û→ u inW 1,p(Ω) and τ̂ → τ and f̃ → f in Lq(Ω;RN) to obtain liminf ε→0 ∫ Ω L( x ε ,∇uε ,τε + f )dx≥ ∫ Ω Lhom ( ∇u(x),τ(x)+ f (x) ) dx. This completes the proof.  111 Before proceeding to the next subsection, we note the following slight exten- sion of Lemma 7.11, which will be needed for Proposition 7.15 below. We note that the proof is known whenGε is independent of the second variable, and here we show that the same proof with minor modification works for general Lagrangians just as in Theorem 7.9. Lemma 7.13 Let Gε and Ghom be as in Theorem 7.9. Then, for any (u,τ) ∈ W 1,p(Ω)×Lq(Ω;RN), there exist a sequence (uε ,τε) such that u−uε ⇀ 0 weakly in W 1,p(Ω) and τε → τ in the T -topology. Furthermore, u− uε ∈W 1,p0 (Ω) and for this sequence: limsup ε→0 Gε(uε ,τε)≤ G(u,τ). Proof. From Theorem 7.9, there exist a sequence (ũε ,τε) with ũε → u in Lp(Ω) and τε → τ in the T -topology, such that Ghom(u,τ) = lim ε→0 Gε(ũε ,τε). Up to a subsequence one may assume that ũε ⇀ u weakly inW 1,p(Ω). Pick any ϕ ∈W 1,p0 (Ω) with ϕ > 0 in Ω. Define: uε(x) :=  ũε u(x)−ϕ(x)≤ ũε ≤ u(x)+ϕ(x) u(x)−ϕ(x) ũε(x)< u(x)−ϕ(x) u(x)+ϕ(x) u(x)+ϕ(x)< ũε(x) . Note that uε−u∈W 1,p0 (Ω) and since ũε ⇀ uweakly inW 1,p(Ω), so uε ⇀ uweakly inW 1,p(Ω). Note that L+C0 ≥ 0. For any f ∈ Lq(Ω;RN) with div( f ) = 0 we have Gε(uε ,τε)+C0|Ω| ≤ ∫ {uε 6=ũε} [ L ( x ε ,∇uε(x), f (x)+ τε(x) ) +C0 ] dx + ∫ {uε=ũε} [ L ( x ε ,∇uε(x), f (x)+ τε(x) ) +C0 ] dx. 112 For x in the set {uε 6= ũε}, the norm of ∇ũε(x) is controlled by the norm of |∇u(x)|+ |∇ϕ(x)|. It follows that Gε(uε ,τε)+C0|Ω| ≤ ∫ {uε 6=ũε} [ C1 ( (|∇u(x)|+ |∇ϕ(x)|)p+ |τε(x)+ f (x)|q+1 ) +C0 ] dx + ∫ Ω [ L ( x ε ,∇ũε(x), f (x)+ τε(x) ) +C0 ] dx. Take now the infimum over all f ∈ Lq(Ω;RN) with div( f ) = 0 and subtract the latter byC0|Ω|. Since |{uε 6= ũε}|→ 0 and Ghom(u,τ) = limε Gε(ũε ,τε), we obtain limsup ε→0 Gε(uε ,τε)≤ G(u,τ). 7.3.2 Variational homogenization of maximal monotone operators on W 1,p0 (Ω) In this section we establish a homogenization result for selfdual Lagrangians on W 1,p0 (Ω)×W−1,q(Ω) where 1p + 1q = 1 and then proceed to prove Theorem 7.1. Theorem 7.14 Let Ω be a regular bounded domain, Q be an open non-degenerate parallelogram in Rn, and L : Ω×RN ×RN → R be an Ω-dependent selfdual La- grangian such that: (1) For each a,b ∈ RN the function x→ L(x,a,b) is Q-periodic, (2) For some constants C0,C1 ≥ 0, we have for every x ∈ RN , C0(|a|p+ |b|q)≤ L(x,a,b)≤C1(|a|p+ |b|q+1), (7.25) where p > 1 and 1p + 1 q = 1. Let u ∗ n → u∗ strongly in W−1,q(Ω) and let un be solutions and τn be momenta for the Dirichlet boundary value problems τn(x) ∈ ∂̄L( xεn ,∇un(x)) a.e. x ∈Ω −div(τn(x)) = u∗n(x) x ∈Ω un ∈W 1,p0 (Ω). (7.26) 113 Then, up to a subsequence, un → u weakly in W 1,p0 (Ω) and τn → τ weakly in Lq(Ω;RN), where u is a solution and τ is a momentum of the homogenized problem τ(x) ∈ ∂̄Lhom(∇u(x)) a.e. x ∈Ω −div(τ(x)) = u∗(x) x ∈Ω u ∈W 1,p0 (Ω), (7.27) where Lhom is the selfdual Lagrangian on RN×RN defined by Lhom(a,b) := min ϕ∈W 1,p# (Q) g∈Lq#(Q;RN) 1 |Q| ∫ Q L ( x,a+Dϕ(x),b+g(x) ) dx. (7.28) This will follow from the following proposition. Proposition 7.15 LetΩ,Q and L be as in Theorem 7.14, and let {Fε ;ε > 0} be the family of selfdual Lagrangians on W 1,p0 (Ω)×W−1,q(Ω) defined by Fε(u,u∗) := inf f∈Lq(Ω;RN) −div f=u∗ ∫ Ω L ( x ε ,∇u(x), f (x) ) dx. Then, there exists a selfdual Lagrangian Fhom on RN ×RN that is a Γ-limit of {Fε ;ε > 0} on W 1,p0 (Ω)×W−1,q(Ω). It is given by the formula Fhom(u,u∗) := inf f∈Lq(Ω;RN) −div f=u∗ ∫ Ω Lhom ( ∇u(x), f (x) ) dx, where Lhom is the selfdual Lagrangian on RN ×RN defined by (7.28), and which satisfies for all (a,b) ∈ RN×RN C0(|a|p+ |b|q−1)≤ Lhom(a,b)≤C1(|a|p+ |b|q+1). Proof. Note first that the selfduality and uniform bounds of Lhom follow from Proposition 7.7. It also follows from Proposition 3.22 that both Fε and Fhom are self- 114 dual Lagrangians onW 1,p0 (Ω)×W−1,q(Ω). Given (u,u∗) ∈W 1,p0 (Ω)×W−1,q(Ω), we now show the existence of a sequence {(uε ,u∗ε) ∈W 1,p0 (Ω)×W−1,q(Ω)} with uε → u weakly inW 1,p0 (Ω) and u∗ε → u∗ strongly inW−1,q(Ω) and such that limsup ε→0 Fε(uε ,u∗ε)≤ Fhom(u,u∗). (7.29) For that we consider {Gε ;ε > 0} be a family of functionals onW 1,p(Ω)×Lq(Ω;RN) defined by Gε(u,τ) := inf f∈Lq(Ω;RN) div f=0 ∫ Ω L ( x ε ,∇u(x),τ(x)+ f (x) ) dx, and Ghom(u,τ) := inf f∈Lq(Ω;RN) div f=0 ∫ Ω Lhom ( ∇u(x),τ(x)+ f (x) ) dx, Take τ ∈ Lq(Ω;RN) such that div(τ)= u∗. It follows from Lemma 7.11 and Lemma 7.13 that there exists (uε ,τε) ∈W 1,p0 (Ω)×Lq(Ω;RN) such that uε → u strongly in Lp(Ω) and τε → τ strongly in Lq(Ω;RN) and limsup ε→0 Gε(uε ,τε)≤ Ghom(u,τ). The sequence uε is bounded in W 1,p 0 (Ω), so we may assume uε → u weakly in W 1,p0 (Ω). Since τε → τ strongly in Lq(Ω;RN), it follows that u∗ε := div(τε) → div(τ) = u∗ strongly inW−1,q(Ω). Thus, the inequality (7.29) follows by noticing that Gε(uε ,τε) = Fε(uε ,u∗ε) and Ghom(u,τ) = Fhom(u,u∗). We shall now show that if (u,u∗) ∈W 1,p0 (Ω)×W−1,q(Ω) and uε → u weakly inW 1,p0 (Ω) and u ∗ ε → u∗ strongly inW−1,q(Ω) then Fhom(u,u∗)≤ liminf ε→0 Fε(uε ,u∗ε). (7.30) Take an arbitrary element in (v,v∗) ∈W 1,p0 (Ω)×W−1,q(Ω). From the above, there exists (vε ,v∗ε) ∈W 1,p0 (Ω)×W−1,q(Ω) with ṽε → v weakly inW 1,p0 (Ω) and v∗ε → v∗ 115 strongly inW−1,q(Ω) and such that limsup ε→0 Fε(vε ,v∗ε)≤ Fhom(v,v∗). By the self duality of Fε we have Fε(uε ,u∗ε) = F ∗ ε (u ∗ ε ,uε) = sup{〈uε ,w∗〉+ 〈u∗ε ,w〉−Fε(w,w∗) : (w,w∗) ∈W 1,p0 (Ω)×W−1,q(Ω)} ≥ 〈uε ,v∗ε〉+ 〈u∗ε ,vε〉−Fε(vε ,v∗ε), from which we get liminf ε→0 Fε(uε ,u∗ε) ≥ liminfε→0 {〈uε ,v ∗ ε〉+ 〈u∗ε ,vε〉−Fε(vε ,v∗ε)} = 〈u,v∗〉+ 〈u∗,v〉− limsup ε→0 Fε(vε ,v∗ε) ≥ 〈u,v∗〉+ 〈u∗,v〉−Fhom(v,v∗). Since the above holds for an arbitrary (v,v∗) ∈W 1,p0 (Ω)×W−1,q(Ω), we obtain F∗hom(u ∗,u)≤ liminf ε→0 Fε(uε ,u∗ε). Taking into consideration that Fhom is selfdual we obtain Fhom(u,u∗) = F∗hom(u ∗,u)≤ liminf ε→0 Fε(uε ,u∗ε), as desired.  Proof of Theorem 7.14. Since (un,τn) are solutions of (7.26), it follows that 0 = ∫ Ω L( x εn ,∇un(x),τn(x))dx− ∫ Ω 〈∇un(x),τn(x)〉RN dx = ∫ Ω L( x εn ,∇un(x),τn(x))dx− ∫ Ω un(x)u∗n(x)dx. (7.31) Due to the coercivity assumption on L and the strong convergence of u∗n, it follows that the sequence un is bounded inW 1,p 0 (Ω) and τn is bounded in L q(Ω;RN). Thus, 116 up to a subsequence, un → u weakly inW 1,p0 (Ω) and τn → τ weakly in Lq(Ω;RN). We also have div(τn) = u∗n → u∗ = div(τ) strongly in W−1,q(Ω), from which we indeed have τn → τ in the T -topology (introduced in Theorem 7.9). Taking f ∈ Lq(Ω;RN) with div f = 0, it follows from (7.31) that∫ Ω L( x εn ,∇un(x),τn(x))dx = ∫ Ω un(x)u∗n(x)dx = − ∫ Ω un(x)div(τn+ f )dx = ∫ Ω 〈∇un(x),τn+ f 〉RN dx ≤ ∫ Ω L ( x εn ,∇un(x),τn+ f (x) ) dx. This indeed shows that∫ Ω L ( x εn ,∇un(x),τn(x) ) dx= inf f∈Lq(Ω;RN) div f=0 ∫ Ω L ( x εn ,∇un(x),τn(x)+ f (x) ) dx. Let Gεn(v, τ̂) := inf f∈Lq(Ω;RN) div f=0 ∫ Ω L ( x ε ,∇v(x), τ̂(x)+ f (x) ) dx. It then follows that ∫ ΩL ( x εn ,∇un(x),τn(x) ) dx=Gεn(un,τn). Define H :W 1,p 0 (Ω)× Lq(Ω;RN)→ R by H(v, τ̃) = ∫Ω〈∇v(x), τ̃(x)〉RN dx. Note that H is continuous if we consider the weak topology ofW 1,p0 (Ω) and the T -topopogy for L q(Ω;RN). It then follows from Lemma 7.12 that∫ Ω Lhom ( ∇u(x),τ(x) ) dx−H(u,τ) ≤ liminf εn→0 [ Gεn(un,τn)−H(un,τn) ] = liminf εn→0 [∫ Ω L( x εn ,∇un(x),τn(x))dx−∫ Ω un(x)div(τn(x))dx ] = 0. 117 On the other hand, we have that∫ Ω Lhom ( ∇u(x),τ(x) ) dx−H(u,τ) =∫ Ω [ Lhom ( ∇u(x),τ(x) )−〈∇u(x),τ(x)〉RN]dx ≥ 0. which means that the latter is indeed zero, i.e.,∫ Ω [ Lhom ( ∇u(x),τ(x) )−〈∇u(x),τ(x)〉RN ]dx= 0. Since the integrand is itself non-negative we have Lhom ( ∇u(x),τ(x) )−〈∇u(x),τ(x)〉RN = 0 a.e. x ∈Ω, which together with −div(τ(x)) = u∗(x), yields τ(x) ∈ ∂̄Lhom(∇u(x)), a.e.x ∈Ω, −div(τ(x)) = u∗(x), x ∈Ω, u ∈W 1,p0 (Ω).  118 Bibliography [AB93] Hedy Attouch and Gerald Beer. On the convergence of subdifferentials of convex functions. Arch. Math., 60:389–400, 1993. [Att84] Hedy Attouch. Variational convergence for functions and operators. Applicable Mathematics Series. Pitman (Advanced Publishing Program), 1984. [Bar84] V. Barbu. 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