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A self-dual approach to stochastic partial differential equations Shirin, Boroushaki 2018

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A self-dual approach to stochasticpartial differential equationsbyShirin BoroushakiB.Sc., Ferdowsi University of Mashhad, 2010M.Sc., Sharif University of Technology, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)January 2018c© Shirin Boroushaki 2018AbstractIn the first part of this thesis, we use the theory of self-duality to providea variational approach for the resolution of a number of stochastic partialdifferential equations. We will be able to address the problem of existenceof solutions to a class of semilinear stochastic partial differential equationsin the form {du+A(t, u(t))dt = B(t, u(t))dWu(0) = u0,(1)where for every t ∈ [0, T ], A(t, ·) is a maximal monotone operator on areflexive Banach space V , and B is a linear or non-linear operator withvalues in a Hilbert space H. We use the fact that any maximal monotoneoperator A can be expressed as a potential of a self-dual Lagrangian Lto associate to the equation (1) a (completely) self-dual functional whoseminimizer on a suitable path space yields a solution.One particular case of (1) which already contains a large number ofstochastic PDEs is when A is the subdifferential of a convex function ϕ.More generally, we can deal with equations of the form{du(t) = −∂ϕ(t, u(t))dt+B(t, u(t))dW (t)u(0) = u0.We also prove the existence of solutions to SPDEs in divergence form in-volving a maximal monotone operator β on Rn, which is not necessarily thegradient of a convex function,{du = div(β(∇u(t, x)))dt+B(t)dW (t) in [0, T ]×Du(0, x) = u0 on ∂D,where D ⊂ Rn is a bounded domain.In the second part of the thesis, we use methods from optimal trans-port to address functional inequalities on the n-dimensional sphere Sn. Weprove Energy-Entropy duality formulas that yield and improve the cele-brated Moser-Onofri inequalities on S2.iiLay SummaryThe evolution of many physical systems can often be described by determin-istic differential equations. However, in many models of natural phenomenathere are uncertainties that have a considerable effect on the evolution ofthe systems under study. These naturally appear while studying populationand genetic dynamics, neurophysiology, financial markets and turbulencein fluid dynamics. The presence of randomness translates into models de-scribed by stochastic differential equations expressing the presence of whitenoise. While variational principles are natural and well developed for thestudy of deterministic models, their use for stochastic equations have beenlimited by the complexity of the differential structure of Brownian motion.In this thesis, we show that a probabilistic version of the recently developedself-dual variational calculus is as efficient in handling stochastic models asthe non-probabilistic counterpart was in dealing with deterministic equa-tions.iiiPrefaceMuch of this dissertation is adapted from two of the author’s research pa-pers: [1] and [9]. In particular, Chapters 4 and 5, which give a variationalresolution of SPDEs, form the main content of [9], A self-dual variationalapproach to stochastic partial differential equations. Chapter 6 is in accor-dance with [1] (joint work with Dr. Ghoussoub and Dr. Agueh), which waspublished in Annales de la Faculte´ des Sciences de Toulouse, Se´r. 6, 26 no2 (2017) p. 217-233. The second manuscript [9] is a joint work with Dr.Ghoussoub. It has been posted on arXiv and is submitted for publication.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1I Self-dual variational principles for stochastic partial dif-ferential equations . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Basics of stochastic calculus . . . . . . . . . . . . . . . . . . . 82.1 Random variables and probability space . . . . . . . . . . . . 82.2 Gaussian measures in Banach spaces . . . . . . . . . . . . . . 102.2.1 Gaussian measures on Hilbert spaces . . . . . . . . . 102.3 Stochastic processes . . . . . . . . . . . . . . . . . . . . . . . 112.3.1 Processes with filtration . . . . . . . . . . . . . . . . 122.4 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Wiener processes . . . . . . . . . . . . . . . . . . . . . . . . . 142.6 Stochasic integral . . . . . . . . . . . . . . . . . . . . . . . . 152.6.1 Operator-valued random variables . . . . . . . . . . . 152.6.2 Construction of the stochastic integral . . . . . . . . 162.7 Stochastic Fubini theorem . . . . . . . . . . . . . . . . . . . 192.8 Itoˆ’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Self-dual Lagrangians and their variational principle . . . 213.1 Basic tools of convex analysis . . . . . . . . . . . . . . . . . . 213.1.1 Subdifferentiability of convex functions . . . . . . . . 22vTable of Contents3.1.2 Legendre duality . . . . . . . . . . . . . . . . . . . . . 233.1.3 Legendre transform of integral functionals . . . . . . 233.2 The class of self-dual Lagrangians . . . . . . . . . . . . . . . 243.3 Self-dual vector fields . . . . . . . . . . . . . . . . . . . . . . 263.4 Variational principle for self-dual functionals . . . . . . . . . 303.4.1 Evolution triples and self-dual Lagrangians . . . . . . 303.4.2 Primal and dual convex optimization problem . . . . 313.5 The class of antisymmetric Hamiltonians . . . . . . . . . . . 343.5.1 Variational principle for self-dual functionals . . . . . 354 Self-dual variational principle for stochastic partial differen-tial equations with additive noise . . . . . . . . . . . . . . . . 404.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2 Lifting random self-dual Lagrangians to Itoˆ path spaces . . . 424.2.1 Self-dual Lagrangians associated to progressively mea-surable monotone fields . . . . . . . . . . . . . . . . 444.2.2 Itoˆ path spaces over a Hilbert space . . . . . . . . . . 454.2.3 Self-dual Lagrangians on Itoˆ spaces of random pro-cesses . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3 Variational resolution of stochastic equations driven by addi-tive noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3.1 A variational principle on Itoˆ space . . . . . . . . . . 494.3.2 Regularization via inf-involution . . . . . . . . . . . . 514.4 Applications to various SPDEs with additive noise . . . . . . 554.4.1 Stochastic evolution driven by diffusion and transport 554.4.2 Stochastic porous media . . . . . . . . . . . . . . . . 564.4.3 Stochastic PDE involving the p-Laplacian . . . . . . 575 Self-dual variational principle for stochastic partial differen-tial equations with non-additive noise . . . . . . . . . . . . . 595.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Non-additive noise driven by self-dual Lagrangians . . . . . . 605.2.1 Stochastic elliptic regularization . . . . . . . . . . . . 625.2.2 A general existence result . . . . . . . . . . . . . . . . 685.3 Non-additive noise driven by monotone vector fields . . . . . 695.3.1 Non-additive noise driven by gradient of convex ener-gies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.3.2 Non-additive noise driven by general monotone vectorfields . . . . . . . . . . . . . . . . . . . . . . . . . . . 71viTable of Contents5.3.3 Non-additive noise driven by monotone vector fieldsin divergence form . . . . . . . . . . . . . . . . . . . . 72II Euclidean Moser-Onofri inequality and its extensionsto higher dimension . . . . . . . . . . . . . . . . . . . . . . . . . 776 A dual Moser-Onofri inequality and its extensions to higherdimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.3 Euclidean n-dimensional Onofri inequality: A duality formula 85Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93viiAcknowledgementsFirst and foremost, I would like to express my sincere gratitude to my su-pervisor, Professor Nassif Ghoussoub whose expertise, understanding andinvaluable guidance and support added considerably to my graduate expe-rience. I thank him for being always available to discuss research problemsand answer my never-ending questions.I would also like to express my gratitude to Dr. Martial Agueh, who,although no longer with us, generously assisted me through our collaborationand writing of my first research paper.I would also like to thank the members of my supervisory committee,Professor Young-Heon Kim and Professor Jun-cheng Wei for their insightfuland instructive comments.Finally, I wish to thank my family for their endless love, support andencouragement throughout my life, and for giving me strength to realize mypotential.viiiChapter 1IntroductionThis thesis is comprised of two independent parts. Both are based on recentadvances in infinite dimensional variational methods: The theory of optimalmass transport, and the self-dual variational calculus.A standard approach to solve several classes of partial differential sys-tems is to represent them as Euler-Lagrange equations whose solutions arecharacterized as critical points to the corresponding functionals of the formI(u) =∫D F (x, u(x),∇u(x))dx, where D is an open bounded subset of Rn.However, there are a large number of partial differential equations involv-ing non-linear, non-local or non self-adjoint operators which do not fall inthe Euler-Lagrange framework. Self-dual variational calculus was developedin the last fifteen years in an effort to construct solutions to such non-variational partial differential equations and evolutions. We refer to themonograph [29] for a comprehensive account of that theory. In the firstpart, which is indeed the core of this thesis, we show how such a calculuscan be applied to solve stochastic partial differential equations, which alsodo not fit in Euler-Lagrange theory, since their solutions are not known tobe critical points of energy functionals. We show here that at least for someof these equations, solutions can be obtained as minima of suitable self-dualfunctionals on Itoˆ spaces of random paths.The genesis of self-dual variational calculus can be traced to a 1970 paperof Brezis-Ekeland [12, 13] (see also Nayroles [40, 41]), where they proposeda variational principle for the heat equation and other gradient flows forconvex energies. The method consists of minimizing the functionalI(u) = ϕ(u) + ϕ∗(−Au) + 〈u,Au〉,where A : V → V ∗ is a non self-adjoint or non-potential operator on theBanach space V , and ϕ is a convex functional on V with ϕ∗ its Legendredual. The basic property of Legendre duality ϕ(u) + ϕ∗(p) ≥ 〈u, p〉 andϕ(u) + ϕ∗(p) = 〈u, p〉 iff p ∈ ∂ϕ(u),yield that infu∈V I(u) ≥ 0. Therefore, the observation is that if the value ofthe infimum is 0 and is attained at some u¯ ∈ V , then the limiting case of1Chapter 1. IntroductionLegendre duality implies that −Au¯ ∈ ∂ϕ(u¯), and hence u¯ is a solution. How-ever, the main difficulty is to prove that the infimum is actually zero. Theconjecture was eventually verified by Ghoussoub-Tzou [34], who identifiedand exploited the self-dual nature of the Lagrangians involved. Since then,the theory was developed in many directions [26, 27, 31], so as to provide ex-istence results for several stationary and parabolic -but so far deterministic-PDEs, which may or may not be Euler-Lagrange equations.While in most examples where the approach was used, the self-dualLagrangians were explicit, an important development in the theory was therealization [30] that in a prior work, Fitzpatrick [23] had associated a (some-what) self-dual Lagrangian to any given monotone vector field. That meantthat the variational theory could apply to any equation involving such op-erators. We refer to the monograph [29] for a survey and for applicationsto existence results for solutions of several PDEs and evolution equations.We also note that since the appearance of this monograph, the theory hasbeen successfully applied to the homogenization of periodic non-self adjointproblems (Ghoussoub-Moameni-Zarate [32]).One of the most important classes of evolutionary equations is the classof (semi-linear) stochastic partial differential equations which are used tomodel and describe many kinds of dynamics in the natural sciences, as wellas in finance. These equations are extensions of Itoˆ stochastic equationsintroduced in the 1940s by Itoˆ. Basic theoretical questions on existence anduniqueness of solutions were asked and answered, under various sets of con-ditions, in the 1970s and 1980s and are still of great interest today. There arebasically three approaches to analyzing SPDEs: the ”martingale approach”[53], the ”semi-group (mild solution) approach” [16] and the ”variationalapproach” [45, 48]. There is an enormously rich literature on all three ap-proaches which cannot be listed here.In this thesis, we follow the ”variational approach” which was initiatedin the celebrated thesis of Pardoux [44], and many other subsequent works[45, 46, 48]. However, the application of self-dual variational method tosolving SPDEs is long overdue, though V. Barbu [6, 7] did use a Brezis-Ekeland approach to address SPDEs driven by gradients of a convex functionand additive noise. We shall deal here with more general situations thatcannot be reduced to the deterministic case. We note that the equationsstudied here have already been solved by other methods, and this work isabout presenting a new variational approach, hoping it will lead to progresson other unresolved equations.In Chapter 2, we present the basics of probability theory in Hilbert andBanach spaces. We list some commonly used concepts from the theory2Chapter 1. Introductionof stochastic processes and stochastic integration with respect to generalHilbert-valued Wiener processes. In Chapter 3, we provide the basic toolsof convex analysis required for the foundation of self-dual systems in termsof the class of self-dual Lagrangians and their corresponding vector fields.We then state the variational principles for minimization of self-dual func-tionals, which also involve the Hamiltonian formulation for the minimizationof direct sum of self-dual functionals.In Chapter 4, we shall tackle basic SPDEs involving additive noise,namely {du(t) = −A(t, u(t))dt+B(t)dW (t)u(0) = u0,(1.1)where u0 ∈ L2(Ω,F0,P;H) for the Hilbert space H, W (t) is a real-valuedWiener process on a complete probability space (Ω,F ,P) with normal fil-tration (Ft)t, and where B : [0, T ]×Ω→ H is a given Hilbert-space valuedprogressively measurable process. Here, A : Ω × [0, T ] × V → 2V ∗ can bea time-dependent adapted –possibly set-valued– maximal monotone map,where V is a Banach space such that V ⊂ H ⊂ V ∗ constitute a Gelfandtriple. The simplest example is where the monotone operator A is givenby the gradient ∂ϕ of a (possibly random and progressively measurable)function ϕ : [0, T ] × H → R ∪ {+∞} such that for every t ∈ [0, T ], thefunction ϕ(t, ·) is convex and lower semi-continuous on a Hilbert space H,and the stochastics is driven by a given progressively measurable additivenoise coefficient B : Ω× [0, T ]→ H. The equation becomes{du(t) = −∂ϕ(t, u(t))dt+B(t)dW (t)u(0) = u0.(1.2)We consider the following Itoˆ space over H,A2H ={u : ΩT → H; u(t) = u(0) +∫ t0u˜(s)ds+∫ t0Fu(s)dW (s)},where u(0) ∈ L2(Ω,F0,P;H), u˜ ∈ L2(ΩT ;H) and Fu ∈ L2(ΩT ;H), whereΩT = Ω × [0, T ]. Here, both the drift u˜ and the diffusive term Fu are pro-gressively measurable.The key idea is that a solution for (1.2) can be obtained by minimizingthe following functional on A2H ,I(u) = E{∫ T0(Lϕ(u(t),−u˜(t))+12MB(Fu(t),−Fu(t)))dt+`u0(u(0), u(T ))},where3Chapter 1. Introduction1. Lϕ is the (possibly random) time-dependent Lagrangian on H × Hgiven byLϕ(u, p) = ϕ(w, t, u) + ϕ∗(w, t, p), (1.3)where ϕ∗ is the Legendre transform of ϕ;2. `u0 is the time-boundary random Lagrangian on H ×H given by`u0(a, b) := `u0(w)(a, b) =12‖a‖2H +12‖b‖2H − 2〈u0(w), a〉H + ‖u0(w)‖2H ;(1.4)3. MB is the random time-dependent diffusive Lagrangian on H × H,given byMB(G1, G2) := ΨB(w,t)(G1) + Ψ∗B(w,t)(G2), (1.5)where ΨB(w,t) : H → R ∪ {+∞} is the convex function ΨB(w,t)(G) =12‖G− 2B(w, t)‖2H .However, it is not sufficient that I attains its infimum on A2H at some v, butone needs to also show that the infimum is actually equal to zero. By usingItoˆ’s formula, we can rewrite I(v) as the sum of 3 non-negative terms0 = I(v) = E∫ T0(ϕ(t, v) + ϕ∗(t,−v˜(t)) + 〈v(t), v˜(t)〉)dt+ 2E∫ T0‖Fv −B‖2H dt+ E ‖v(0)− u0‖2H ,which yields that for almost all t ∈ [0, T ], P-a.s.−v˜(t) ∈ ∂ϕ(v(t)), B = Fv, v(0) = u0.In other words, v(t) = u0−∫ t0 ∂ϕ(s, v(s))ds+∫ t0 B(s)dW (s), where the laststochastic integral is in the sense of Itoˆ. Hence, v is a solution to (1.2).The main focus of Chapter 4 is to extend this argument to more generalLagrangians L and consequently resolve Equation (1.1). We then presentsome applications of the method to classical SPDEs such as stochastic evolu-tion driven by a diffusion and a transport operator, stochastic porous mediaequation, and quasi-linear equations involving p-Laplacian.In Chapter 5, we will deal with SPDEs driven by monotone vector fieldsand involving a non-additive noise. These can take the form{du(t) = −A(t, u(t))dt+B(t, u(t))dW (t)u(0) = u0,(1.6)4Chapter 1. Introductionwhere u → B(t, u) is now a progressively measurable linear or non-linearoperator. Analogously, we consider the Itoˆ space this time over the reflexiveBanach space V to find a solution that is V -valued. To this end, we shallstrengthen the norm on the Itoˆ space over a Gelfand triple, at the cost oflosing coercivity, that we shall recover through perturbation methods. Bysolutions, we mean progressively measurable processes u, valued in suitableSobolev spaces, that verify the integral equationu(t) = u0 −∫ t0A(s, u(s))ds+∫ t0B(s, u(s))dW (s).To variationally resolve Equation (1.6), we shall use the self-dual ap-proach regarding minimization of direct sum of self-dual functionals (seeTheorem 3.5) together with the so-called technique of elliptic regularizationto ensure coercivity. We will then let the perturbations go to zero to recoverthe equation and conclude the existence of a solution. We then apply thisresult to resolve an equation of the form (1.6) and more generally,{du = div(β(∇u(t, x)))dt+B(u(t))dW (t) in [0, T ]×Du(0, x) = u0(x) on ∂D,where D is a bounded domain in Rn, the initial position u0 belongs toL2(Ω,F0,P;L2(D)), and the vector field β is a progressively measurablemaximal monotone operator on Rn (see Subsection 5.3.3).In part (II) of this thesis, we use optimal mass transport to provide anew proof and a dual formula to the Moser-Onofri inequality on S2,14∫S2|∇u|2 dω +∫S2udω ≥ log(∫S2eu dω).This is in the same spirit as the approach of Cordero-Erausquin, Nazaretand Villani [15] to the Sobolev and Gagliardo-Nirenberg inequalities andthe one of Agueh-Ghoussoub-Kang [2] to more general settings. We usean equivalent formulation of the Onofri inequality which is obtained viastereographic projection of (6.1), namely116pi∫R2|∇u|2 dx+∫R2u dµ2 ≥ log(∫R2eu dµ2)∀u ∈ H1(R2).However, once a projection on R2 is performed, several new hurdles ap-pear that are not the case in [15] or [2]: Functions are not necessarily5Chapter 1. Introductionof compact support, hence boundary terms need to be evaluated. More-over, the corresponding dual free energy of the reference probability densityµ2(x) =1pi(1+|x|2)2 is not finite on the whole space, which requires the in-troduction of a renormalized free energy into the dual formula. We alsoextend this duality to higher dimensions and establish an extension of theOnofri inequality to spheres Sn with n ≥ 2. What is remarkable is thatthe corresponding free energy is again given by F (ρ) = −nρ1− 1n , whichmeans that both the prescribed scalar curvature problem and the prescribedGaussian curvature problem lead essentially to the same dual problem whoseextremals are stationary solutions of the fast diffusion equations.6Part ISelf-dual variationalprinciples for stochasticpartial differential equations7Chapter 2Basics of stochastic calculusThe prime objective of this chapter is to make this dissertation self-containedin terms of definitions and facts from the measure theoretic foundations ofprobability theory. A fundamental element in stochastic differential equa-tions is the presence of the diffusion term which in our case corresponds tothe stochastic integral with respect to a given Wiener process. Therefore,the goal in this chapter is to provide the reader with sufficient material todevelop the probability theory in infinite dimension, namely Hilbert-valuedstochastic processes and the construction of stochastic integrals as processeswith values in these Hilbert spaces. Our main source for most of the com-ponents of this chapter is [16] and [48].2.1 Random variables and probability spaceA measurable space is a pair (Ω,F) where Ω is a non-empty set and Fis a σ-field, also called a σ-algebra, of subsets of Ω. This means that thefamily F contains the set Ω and is closed under the operation of takingcomplements and countable unions of its elements. If (Ω,F) and (V,G) aretwo measurable spaces, then a mapping X from Ω into V such that forarbitrary A ∈ G, the set X−1(A) = {ω ∈ Ω;X(ω) ∈ A} =: {X ∈ A} belongsto F is called is called a measurable mapping or a random variable from(Ω,F) into (V,G) or a V -valued random variable.Assume that V is a metric space, then the Borel σ-field of V is thesmallest σ-field containing all closed (or open) subsets of V and it will bedenoted as B(V ).In fact, if V is a separable Banach space, and V ∗ is its topological dual,then a mapping X : Ω→ V is a V -valued random variable if and only if forarbitrary ζ ∈ V ∗, ζ(X) : Ω→ R is an R-valued random variable.Definition 2.1. A probability measure on a measurable space (Ω,F) is aσ-additive function P from F into [0, 1] such that P(Ω) = 1. The triplet(Ω,F ,P) is called a probability space.82.1. Random variables and probability spaceIf the triplet (Ω,F ,P) is a probability space, we setF¯ = {A ⊂ Ω; ∃ B,C ∈ F ;B ⊂ A ⊂ C, P(B) = P(C)}.F¯ is a σ-field, called the completion of F . If F = F¯ , then the probabilityspace (Ω,F ,P) is said to be complete. Equivalently, (Ω,F ,P) is complete ifany subset N of B ∈ F with P(B) = 0 is also in F .If X is a random variable from (Ω,F) into (V,G) and P a probabilitymeasure on Ω, then by L(X) we denote the image of P by the mapping X;L(X)(A) = P(ω ∈ Ω;X(ω) ∈ A)= P({X ∈ A}) ∀ A ∈ GThe measure µ = L(X) is called the distribution or the law of X.Similar to the definition of Lebesgue integral of real-valued functions, onecan define the integral for a real-valued random variable and analogously theintegral of a random variable with values in a separable Banach space V . Infact, the random variable X is said to be Bochner integrable or just integrableif ∫Ω‖X(ω)‖P(dω) <∞.The integral∫ΩXdP will often be denoted by E(X), the expectation ofX and it has many properties of the Lebesgue integral.Finally, in the last part of this section we introduce some basic functionspaces. We denote by L1(Ω,F ,P;V ) the set of all equivalence classes ofV -valued random variables (with respect to the equivalence relation X ∼Y ⇔ X = Y a.s. ). One can check that L1(Ω,F ,P;V ), equipped with thenorm‖X‖1 = E(‖X‖V )is a Banach space. In a similar way, one can define Lp(Ω,F ,P;V ), forarbitrary p > 1 with the norm‖X‖p = (E‖X‖pV )1/p.92.2. Gaussian measures in Banach spacesIf X,Y ∈ L2(Ω,F ,P;H) and H is a Hilbert space with inner product〈·, ·〉, we define the covariance operator of X by the formulaCov(X) = E((X − E(X))⊗ (X − E(X)))Cov(X) is a symmetric positive and nuclear operator andTr[Cov(X)] = E(|X − E(X)|2)2.2 Gaussian measures in Banach spacesDefinition 2.2. A Gaussian measure µ on R is either concentrated at onepoint µ = δm or has a density1√2piqe− 12q(x−m)2for x ∈ R,where q > 0 and m ∈ R are respectively the variance and the mean of therandom variable. Such a measure is denoted by N (m, q).The Gaussian measure µ on Rn has the density1√(2pi)n detQe−12〈Q−1(x−m),x−m〉 for x ∈ Rn,where Q is a positive-definite symmetric n × n matrix, known as thecovariance matrix, and m ∈ Rn is the mean, and this measure also is denotedby N (m,Q).Now let V be a separable Banach space. A probability measure µ on(V,B(V )) is said to be a Gaussian measure if and only if the law of anarbitrary linear functional h ∈ V ∗, considered as a random variable on(V,B(V ), µ), is a Gaussian measure on (R,B(R)). If the law of each h ∈ V ∗is in addition a symmetric (zero mean) Gaussian law on R then µ is calleda symmetric Gaussian measure.2.2.1 Gaussian measures on Hilbert spacesFor Gaussian measures on Hilbert spaces more precise information can begiven. Let H be a Hilbert space, according to the general definition onBanach spaces, a probability measure µ on (H,B(H)) is called Gaussian iffor arbitrary h ∈ H there exist m ∈ R and q ≥ 0 such that,102.3. Stochastic processesµ({x ∈ H; 〈h, x〉 ∈ A})= N (m, q)(A), ∀A ∈ B(R).It is proved that for a Gaussian measure µ on (H,B(H)) there exist anelement m ∈ H and a linear operator Q, such that∫H〈h, x〉µ(dx) = 〈m,h〉, ∀h ∈ H,∫H〈h1, x−m〉〈h2, x−m〉µ(dx) = 〈Qh1, h2〉, ∀h1, h2 ∈ H.The vector m is called the mean and Q is called the covariance operatorof µ. It is clear that the operator Q is symmetric and also non-negative.The Gaussian measure is denoted by N (m,Q).The following proposition shows that the covariance operator is nuclear.A proof can be found in [16].Proposition 2.1. Let µ be a Gaussian probability measure with mean 0 andcovariance Q. Then Q has finite trace.2.3 Stochastic processesAssume that V is a separable Banach space and let B(V ) be the σ-field of itsBorel subsets. Let (Ω,F ,P) be a probability space, and let I be an intervalof R. An arbitrary family X = {X(t)}t∈I , of V -valued random variablesX(t), t ∈ I defined on Ω is called a stochastic process. We also say thatX(t) is a stochastic process on I. We set X(t, ω) = X(t)(ω) for all t ∈ Iand ω ∈ Ω. Functions X(·, ω) are called the trajectories of X(t).A stochastic process Y is called a modification or a version of X ifP(ω ∈ Ω;X(t, ω) 6= Y (t, ω)) = 0 ∀ t ∈ I.A process X is called measurable if the mapping X(·, ·) : I × Ω → V isB(I)⊗F-measurable.X is called stochastically continuous at t0 ∈ I if for all  > 0 and allδ > 0 there exists ρ > 0 such thatP(‖X(t)−X(0)‖ ≥ ) ≤ δ ∀ t ∈ [t0 − ρ, t0 + ρ] ∩ [0, T ].112.3. Stochastic processes2.3.1 Processes with filtrationDefinition 2.3. Assume that I = [0, T ] or [0,+∞]. A filtration on theprobability space (Ω,F ,P) is an increasing family of σ-fields {Ft}, t ∈ Isuch that, for all s ≤ t in I, we have Fs ⊂ Ft ⊂ {Ft}t∈I .Denote by Ft+ the intersection of all Fs where s > t. The filtration issaid to be normal if1. F0 contains all A ∈ F such that P(A) = 0.2. Ft = Ft+ =⋂s>tFs for all t ∈ I.If for arbitrary t ∈ I the random variable X(t) is Ft-measurable thenthe process X is said to be adapted (to the family Ft).X is progressively measurable if for every t ∈ [0, T ] the mapping[0, t]× Ω→ V,(s, ω)→ X(s, ω)is B([0, t])⊗Ft-measurable.Proposition 2.2. Let X(t), t ∈ [0, T ] be a stochastically continuous andadapted process with values in a separable Banach space V . Then X has aprogressively measurable version.Definition 2.4. Let P∞ be the σ-field generated by the sets of the form(s, t] × F , where 0 ≤ s < t < ∞, F ∈ Fs, and {0} × F for F ∈ F0. Thisσ-field is called a predictable σ-field and its elements are predictable sets.The restriction of the σ-field P∞ to [0, T ]× Ω is denoted by PT .An arbitrary measurable mapping from ([0,∞] × Ω,P∞) or ([0, T ] ×Ω,PT ) into (V,B(V )) is called a predictable process. A predictable processis necessarily an adapted one. Predictable processes form a large class ofprocesses, as can be seen in the next proposition.Proposition 2.3. Assume that Φ is an adapted and stochastically contin-uous process on an interval [0, T ]. Then the process Φ has a predictableversion on [0, T ].122.4. Martingales2.4 MartingalesLet X(t) be a V -valued process. If E‖X(t)‖V < +∞ for all t ∈ [0, T ] thenthe process is called integrable. An integrable and adapted V -valued processX(t), t ∈ [0, T ] is said to be a martingale if for arbitrary t, s,∈ [0, T ], t ≥ sE(X(t)|Fs) = X(s), P− a.s.For fixed T > 0, we denote by M 2T (V ) the space of all V -valued contin-uous, square integrable martingales M , such that M(0) = 0.Proposition 2.4. The space M 2T (V ) equipped with the norm‖M‖M 2T (V ) =(E supt∈[0,T ]‖M(t)‖2) 12is a Banach space.If M ∈M 2T (R) then there exists a unique increasing predictable process〈〈M(·)〉〉, starting from 0, such that the processM2(t)− 〈〈M(·)〉〉, t ∈ [0, T ]is a continuous martingale. The process 〈〈M(·)〉〉 is called the quadraticvariation of M .To define the quadratic variation process for M ∈ M 2T (H) where H isa separable Hilbert space, denote by L1 := L1(H) the space of all nuclearoperators on H equipped with the nuclear norm. An L1-valued continuous,adapted and increasing process Z such that Z(0) = 0 is said to be a quadraticvariation process of the martingale M(·) if and only if for arbitrary a, b ∈ Hthe process〈M(t), a〉〈M(t), b〉 − 〈Z(t)a, b〉, t ∈ [0, T ]is an {Ft}-martingale. Z(·) is uniquely determined and is also denoted by〈〈M(t)〉〉.Proposition 2.5. An arbitrary M ∈ M 2T (H) has exactly one quadraticvariation process.A non-negative random variable τ defined on (Ω,F) is said to be anFt-stopping time if for arbitrary t ≥ 0, {ω ∈ Ω; τ(ω) ≤ t} ∈ Ft .132.5. Wiener processes2.5 Wiener processesLet U be a Hilbert space with inner product 〈 , 〉U , and let Q be a traceclass non-negative operator on a Hilbert space U .Definition 2.5. A U -valued stochastic process W (t), t ∈ [0, T ] on a proba-bility space (Ω,F ,P) is called a Q-Wiener process ifi. W (0) = 0ii. W has P-a.s. continuous trajectories.iii. W has independent increments.iv. L(W (t)−W (s)) = N (0, (t− s)Q), t ≥ s ≥ 0.Note that there exists a complete orthonormal system {ek} in U and abounded sequence of non-negative real numbers {λk} such thatQek = λkek, k ∈ N.The following proposition shows that the operator Q can completely char-acterize the distribution of W .Proposition 2.6. Assume that W (t) is a Q-Wiener process, then1. W (t) is a Gaussian process on U , andE(W (t)) = 0, Cov(W (t)) = tQ, t ≥ 02. (Representation of the Q-Wiener process)For arbitrary t ≥ 0, W has the expansionW (t) =∑k∈N√λk βk(t) ek =∑k∈Nβk(t)Q12 ek, t ≥ 0 (2.1)whereβk(t) =1√λk〈W (t), ek〉, k ∈ Nare mutually independent real-valued Brownian motions on (Ω,F ,P).The series converges in L2(Ω,F ,P;C([0, T ];U)).Note that the quadratic variation of a Q-Wiener process in U , withTrQ < +∞, is given by 〈〈W (t)〉〉 = tQ, t ≥ 0.142.6. Stochasic integralDefinition 2.6. A Q-Wiener process W (t), t ∈ [0, T ] is called a Wienerprocess with respect to a filtration Ft if• W (t) is adapted to Ft, t ∈ [0, T ] and• W (t)−W (s) is independent of Fs for all 0 ≤ s ≤ t ≤ T .In this thesis, we only consider real-valued Brownian motions, i.e. U =R, and hence Q = I.2.6 Stochasic integral2.6.1 Operator-valued random variablesOne important concept and of great interest is operator-valued randomvariables which are one of the main elements in construction of stochas-tic integrals. Let U and H be two separable Hilbert spaces and denote byL = L(U,H) the set of all linear bounded operators from U into H. Theset L is a linear space and, equipped with the operator norm, becomes aBanach space. However, if both spaces are infinite dimensional, then L isnot a separable space.The lack of separability of L implies also that Bochners definition cannotbe applied directly to the L-valued functions. To overcome these difficulties,one option is restrict our investigation to smaller spaces - the space L1(U,H)of all nuclear operators from U into H, or the space L2(U,H) of all Hilbert-Schmidt operators from U into H, and then the non-measurability problemarisen from non-separability does not appear, that is because these smallerspaces are separable Banach spaces (L2(U,H) is a Hilbert space).A function Φ(·) from Ω into L is said to be strongly measurable if forarbitrary u ∈ U the function Φ(·)u is measurable as a mapping from (Ω,F)into (H,B(H)). Let L be the smallest σ-field of subsets of L containing allsets of the form {Φ ∈ L; for Φu ∈ A}, u ∈ U, A ∈ B(H), then Φ : Ω→ Lis a strongly measurable mapping from (Ω,F) into (L,L). Elements of Lare called strongly measurable.152.6. Stochasic integralHilbert-Schmidt operatorsDefinition 2.7. Let T ∈ L(U) and let {ek}, k ∈ N be an ONB of U . WedefineTrT :=∞∑k=1〈Tek, ek〉Uif the series is convergent.Note that the trace definition above is independent of the choice of theorthonormal basis.Definition 2.8. A bounded linear operator T : U → H is called Hilbert-Schmidt if‖T‖L2(U,H) :=∞∑k=1‖Tek‖2H <∞where {ek} is an ONB of U .We denote the space of all Hilbert-Schmidt operators from U to H byL2(U,H).The space of Hilbert-Schmidt operators forms a separable Hilbert spacewith the inner product defined as the following:For S, T ∈ L2(U,H) and {ek} an ONB of U , we set〈T, S〉L2 :=∞∑k=1〈Tek, Sek〉H .2.6.2 Construction of the stochastic integralWe first state the definition of the stochastic integral for an elementaryprocess, and we then briefly present the steps to extend it to more generalprocesses.We are given W (t) a Q-Wiener process in (Ω,F ,P) having values in theHilbert space U with respect to a normal filtration Ft, t ∈ [0, T ].Definition 2.9. An L = L(U,H)-valued process Φ(t), t ∈ [0, T ] taking onlya finite number of values is said to be elementary if there exists a sequence0 = t0 < t1 < ... < tk = T and a sequence Φ0,Φ1, ...,Φk−1 of L-valuedrandom variables taking on only a finite number of values such that Φm areFtm-measurable andΦ(t) = Φm t ∈ (tm, tm+1], m = 0, 1, ..., k − 1.162.6. Stochasic integralFor elementary processes Φ, one defines the stochastic integral by theformula(Φ·W )(t) :=∫ t0Φ(s)dW (s) =k−1∑m=0Φm(Wtm+1∧t−Wtm∧t), t ∈ [0, T ] (2.2)In the construction of stochastic integral in a general setting, we need towork with the subspace U0 = Q1/2(U) of U which is a Hilbert space endowedwith the inner product〈u, v〉0 =∞∑k=11λk〈u, ek〉〈v, ek〉 = 〈Q−1/2u,Q−1/2v〉, u, v ∈ U0.Furthermore, another key concept in the construction of the stochasticintegral is the space of all Hilbert-Schmidt operators from U0 to H, L02 =L2(U0, H), which is a separable Hilbert space equipped with the norm||Ψ||2L02 =∞∑i,j=1|〈Ψgi, fj〉|2 =∞∑i,j=1λi|〈Ψei, fj〉|2= ||ΨQ1/2||2L2(U,H)= Tr [(ΨQ1/2)(ΨQ1/2)∗]where {gi}, with gi =√λiei, {ei} and {fi} are complete ONB in U0, U , andH respectively.Let Φ(t), t ∈ [0, T ] be a measurable L02-valued process; we define thefollowing norm on Φ by|||Φ|||t =[E∫ t0||Φ(s)||2L02ds]1/2=[E∫ t0Tr [(Φ(s)Q1/2)(Φ(s)Q1/2)∗]ds]1/2Proposition 2.7. If a process Φ is elementary and |||Φ|||T < ∞, then theprocess Φ·W is a continuous, square integrable H-valued martingle on [0, T ],andE|Φ ·W |2 = |||Φ|||2t , 0 ≤ t ≤ T (2.3)172.6. Stochasic integralIn order to take into account the adaptability of the considered processeswe need to regard the class of integrands that are predictable with values inL02, more precisely, the mapping from (Ω× [0, T ],PT ) into (L02,B(L02)).Proposition 2.8. If Φ is an L20-predictable process such that |||Φ|||T <∞, then there exists a sequence {Φn} of elementary processes such that|||Φn − Φ|||T → 0 as n→∞.Consider the setN 2W (0, T ;L02) : = {Φ : Ω× [0, T ]→ L02; Φ is predictable, and |||Φ|||T <∞}= L2(Ω× [0, T ],PT ,P⊗ dt;L20)or in short N 2W . By Propositio 2.8, the set of elementary processes is adense subset of N 2W , thus with Proposition 2.7 we have that the stochasticintegral Φ·W is an isometric transformation from the dense set of elementaryprocesses into M 2T (H), and this is the key fact to extend the integral to allelements of N 2W . The so-called localization procedure provides the final stepto extend the definition of the stochastic integral where one can relax thefinite norm condition to the followingP(∫ T0‖Φ(s)‖2L02ds <∞)= 1.All such processes are called stochastically integrable on [0, T ].For Φ ∈ N 2W , one defines τn := inf{t ∈ [0, T ];∫ t0 ‖Φ(s)‖2L02ds > n} ∧ T ,which is an increasing sequence of stopping times with respect to Ft, t ∈[0, T ], such thatE(∫ T0‖1(0,τn](s)Φ(s)‖2L02ds)≤ n <∞.In addition, the processes 1(0,τn]Φ, n ∈ N are still L02-predictable since 1(0,τn]is left-continuous and Ft-adapted. Thus we get that for t ∈ [0, T ] thestochastic integrals∫ t0 1(0,τn](s)Φ(s)dW (s) are well defined for all n ∈ N,and hence we set∫ t0Φ(s)dW (s) :=∫ t01(0,τn](s)Φ(s)dW (s),where n is an arbitrary natural number such that τn ≥ t.182.7. Stochastic Fubini theoremNow finally the following Lemma shows that for arbitrary m < n andt ∈ [0, T ] on {τm ≥ t} ⊂ {τn ≥ t}, we have∫ t01(0,τn](s)Φ(s)dW (s) =∫ t01(0,τm](s)Φ(s)dW (s), P− a.s.Lemma 2.1. Assume that Φ ∈ N 2W and that τ is an Fτ -stopping time suchthat P(τ ≤ T ) = 1. Then∫ t01(0,τ ](s)Φ(s)dW (s) = Φ ·W (τ ∧ t) =∫ τ∧t0Φ(s)dW (s).2.7 Stochastic Fubini theoremLet (V,ΣV ) be a measurable space and let (ω, t, u)→ Φ(ω, t, u) be a measur-able mapping from (ΩT × V,PT ⊗B(V )) into (L02,B(L02)). In particular, forarbitrary u ∈ V , (·, ·, u) is a predictable L02-valued process. Let in additionµ be a finite positive measure on (V,ΣV ).The following stochastic version of the Fubini theorem will be used inthe sequel.Theorem 2.1. Let Φ described as above also satisfies∫V|||Φ(·, ·, u)|||µ(dx) < +∞,then P-a.s.∫V(∫ T0Φ(t, u)dW (t))µ(dx) =∫ T0(∫VΦ(t, u)µ(dx))dW (t).2.8 Itoˆ’s formulaAssume that Φ is a stochastically integrable L02-valued process in [0, T ], u˜ anH-valued predictable process P-a.s., Bochner integrable on [0, T ], and u(0)a F0-measurable H-valued random variable. Then the following process iswell defined.u(t) = u(0) +∫ t0u˜(s)ds+∫ t0Φ(s)dW (s), t ∈ [0, T ]. (2.4)192.8. Itoˆ’s formulaTheorem 2.2. Assume that a function F : [0, T ] × H → R and its par-tial derivatives Ft, Fx, Fxx are uniformly continuous on bounded subsets of[0, T ]×H, then for all t ∈ [0, T ], P-a.s. the following Itoˆ formula holdsF (t, u(t)) = F (0, u(0)) +∫ t0〈Fx(s, u(s)),Φ(s)dW (s)〉 (2.5)+∫ t0(Ft(s, u(s)) + 〈Fx(s, u(s)), u˜(s)〉)ds+12∫ t0Tr(Fxx(s, u(s))(Φ(s)Q1/2)(Φ(s)Q1/2)∗)ds,where Q ∈ L(U) is the covariance operator corresponding to the Q-Wienerprocess W .In particular if F (u) = ‖u‖2H , then the formula (2.5) for all t ∈ [0, T ],P-a.s. reduces to the following‖u(t)‖2H = ‖u(0)‖2H +∫ t0(2〈u(s), u˜(s)〉H + ‖Φ(s)‖2L2(U,H))ds+ 2∫ t0〈u(s),Φ(s)dW (s)〉H ,and consequentlyE(‖u(t)‖2H) = E(‖u(0)‖2H)+E∫ t0(2〈u(s), u˜(s)〉H+‖Φ(s)‖2L2(U,H))ds. (2.6)Considering the Itoˆ’s formula as (2.6), one can obtain an analogous inte-gration by parts formula for two processes u and v of the form (2.4); namelyif u takes the form (2.4) and also v is given byv(t) = v(0) +∫ t0v˜(s)ds+∫ t0Ψ(s)dW (s), t ∈ [0, T ],then the following holdsE∫ T0〈u(t), v˜(t)〉dt =− E∫ T0〈v(t), u˜(t)〉dt− E∫ T0〈Φ(t),Ψ(t)〉L2(U,H)dt+ E〈u(T ), v(T )〉 − E〈u(0), v(0)〉. (2.7)20Chapter 3Self-dual Lagrangians andtheir variational principleIn this chapter, we state the preliminary material required for variationalframework of self-dual systems. We start by recalling the basic concepts andrelevant tools of convex analysis that will be frequently used in the sequel.The material in the first section of this chapter is quite standard and can befound in most books on convex analysis, such as [49] or [47]. The followingsections on self-dual analysis is mostly taken from [29], which is our mainreference in this direction.In this work, the approach toward stochatic PDEs is based upon convexcalculus on ”phase space”, V ×V ∗, where V is a reflexive Banach space andV ∗ is its topological dual. We shall therefore consider Lagrangians on V ×V ∗that are convex and lower semi-continuous in both variables. All elementsof convex analysis will apply, but the calculus on state space becomes muchricher for many reasons, such as the possibility of introducing associatedHamiltonians, which are in fact the Legendre dual but in one variable. Wealso state the connection of class of self-dual Lagrangians on V × V ∗ withmaximal monotone operators on V in such a way that these operators can beconsidered as potentials to the Lagrangians for which a generalized notionof subdifferential will be defined.The main advantage of self-dual Lagrangians is the fact that various non-variational PDEs can be formulated variationally in the setting of completelyself-dual systems, and a solution can be obtained by minimizing ”self-dualfunctionals” with 0 as their minimal value.3.1 Basic tools of convex analysisDefinition 3.1. A function ϕ : V → R ∪ {+∞} on a Banach space V issaid to be:1. (weakly) lower semi-continuous (l.s.c), if its epigraph Epi(ϕ), is closedfor the (weak) norm topology, or equivalently for every sequence xn in213.1. Basic tools of convex analysisV that converges (weakly) strongly to x, then ϕ(x) ≤ lim infnϕ(xn).2. convex, if Epi(ϕ) is a convex subset of V × R, which is equivalent toϕ(λu+ (1− λ)v) ≤ λϕ(u) + (1− λ)ϕ(v) for u, v ∈ V and λ ∈ R.3. proper if its effective domain is non-empty i.e.Dom(ϕ) = {u ∈ V ; ϕ(u) < +∞} 6= ∅.3.1.1 Subdifferentiability of convex functionsDefinition 3.2. Let ϕ : V → R∪{+∞} be a convex lower semi-continuousfunction on a Banach space V . Define the subdifferential ∂ϕ of ϕ to be thefollowing set-valued function:∂ϕ(u) = {p ∈ V ∗; 〈p, v − u〉 ≤ ϕ(v)− ϕ(u) ∀v ∈ V },and if u /∈ Dom(ϕ), ∂ϕ(u) = ∅.The subdifferential ∂ϕ(u) is a closed convex subset of the dual space V ∗.It can, however, be empty even though u ∈ Dom(ϕ), and we shall writeDom(∂ϕ) = {u ∈ V ; ∂ϕ(u) 6= ∅}.Definition 3.3. A subset G ∈ V × V ∗ is said to be1. monotone, if for every (u, p) and (v, q) in G,〈u− v, p− q〉 ≥ 0.2. maximal monotone, if it is maximal in the family of monotone sub-sets of V × V ∗ ordered by the set inclusion.3. cyclically monotone, provided that for any finite number of points(xi, pi)ni=0 in G with x0 = xn, we haven∑i=1〈pi, xi − xi−1〉 ≥ 0.A set-valued map A : V → 2V ∗ is then said to be monotone (resp., max-imal monotone) provided its graph G(A) = {(u, p) ∈ V × V ∗, p ∈ A(u)} ismonotone (resp., maximal monotone).The following theorem was established by Rockafellar [49].223.1. Basic tools of convex analysisTheorem 3.1. Let ϕ : V → R ∪ {+∞} be a proper convex and lowersemi-continuous function on a Banach space V . Then its differential mapu→ ∂ϕ(u) is a maximal monotone map.Conversely, if A : V → 2V ∗ is a maximal cyclically monotone map witha non-empty domain, then there exists a proper convex and lower semi-continuous functional ϕ on V such that A = ∂ϕ.3.1.2 Legendre dualityDefinition 3.4. For a convex function ϕ : V → R ∪ {+∞}, its Fenchel-Legendre dual, ϕ∗ : V ∗ → R ∪ {+∞} is defined byϕ∗(p) = supu∈V{〈u, p〉 − ϕ(u)}Proposition 3.1. Let ϕ : V → R∪{+∞} be a proper function on a reflexiveBanach space. The following properties then hold:1. ϕ∗ is a proper convex lower semi-continuous function from V ∗ to R ∪{+∞}.2. ϕ∗∗ := (ϕ∗)∗ : V → R ∪ {+∞} is the largest convex lower semi-continuous function below ϕ. Moreover, ϕ = ϕ∗∗ if and only if ϕ isconvex and lower semi-continuous on V .3. For every (u, p) ∈ V × V ∗, we have ϕ(u) + ϕ∗(p) ≥ 〈u, p〉, and thefollowing are equivalent:• ϕ(u) + ϕ∗(p) = 〈u, p〉• p ∈ ∂ϕ(u)• u ∈ ∂ϕ∗(p)For a complete argument on Legendre duality, we refer the interestedreader to [47, 49]3.1.3 Legendre transform of integral functionalsLet D be a Borel subset of Rn with finite Lebesgue measure, and let Vbe a separable reflexive Banach space. Consider a bounded below functionϕ : D × V → R ∪ {+∞} that is measurable with respect to the σ-fieldgenerated by the products of Lebesgue sets in D and Borel sets in V . We233.2. The class of self-dual Lagrangianscan associate to ϕ a functional Φ defined on Lα(D;V ) for α ∈ [1,∞) via theformulaΦ(u) =∫Dϕ(x, u(x))dx, u ∈ Lα(D;V )For x ∈ D,u ∈ V , and p ∈ V ∗, with the obvious notationϕ∗(x, p) = ϕ(x, ·)∗(p), ∂ϕ(x, u) = ∂ϕ(x, ·)(u)We then have the following proposition which describes the relation be-tween ϕ and its ’integral’ Φ, and that how their Legendre transforms andsubdifferentials are related. A proof can be found in [19].Proposition 3.2. Assume V is a reflexive and separable Banach space, that1 ≤ α ≤ +∞ with 1α + 1β = 1 , and that ϕ : D × V → R ∪ {+∞} is jointlymeasurable such that∫D |ϕ∗(x, p¯(x))|dx < ∞ for some p¯ ∈ Lβ(D;V ) whichholds in particular if ϕ is bounded below on D × V .1. If the function ϕ(x, ·) is lower semi-continuous on V for almost everyx ∈ D, then Φ is lower semi-continuous on Lα(D;V ).2. If ϕ(x, ·) is convex on V for almost every x ∈ D, then Φ is convex onLα(D;V ).3. If ϕ(x, ·) is convex and lower semi-continuous on V for almost everyx ∈ D, and if Φ(u¯) < ∞ for some u¯ ∈ L∞(D;V ), then the Legendretransform of Φ on Lβ(D;V ) is given byΦ∗(p) =∫Dϕ∗(x, p(x))dx for all p ∈ Lβ(D;V ).4. If∫D |ϕ(x, u¯(x))|dx < ∞ and∫D |ϕ∗(x, p¯(x))|dx < ∞ for some u¯ andp¯ in L∞(D;V ), then for every u ∈ Lα(D;V ) we have∂Φ(u) = {p ∈ Lβ(D;V ); p(x) ∈ ∂ϕ(x, u(x)) a.e.}.3.2 The class of self-dual LagrangiansLet V be a reflexive Banach space. Functions L : V × V ∗ → R ∪ {+∞}on phase space V × V ∗ will be called Lagrangians. We consider the class ofLagrangians which are proper, convex and lower semi-continuous (l.s.c) inboth variables. For any (q, v) ∈ V ∗×V , the Legendre-Fenchel dual (in bothvariables) of L is defined by243.2. The class of self-dual LagrangiansL∗(q, v) = sup{〈q, u〉+ 〈v, p〉 − L(u, p); u ∈ V, p ∈ V ∗}The (partial) domains of a Lagrangian L are defined asDom1(L) = {u ∈ V ; L(u, p) <∞, for some p ∈ V ∗}andDom2(L) = {p ∈ V ∗; L(u, p) <∞, for some u ∈ V }Remark 3.1. To any pair of proper convex l.s.c functions ϕ and ψ on aBanach space V , one can associate a Lagrangian on state space V × V ∗via the formula L(u, p) = ϕ(u) + ψ∗(p). It’s Legendre transform is thenL∗(p, u) = ψ(u)+ϕ∗(p) with Dom1(L) = Dom(ϕ) and Dom2(L) = Dom(ψ).The following definition represents an important class of Lagrangians onphase space called the self-dual Lagrangians that are the building blocks ofour approach to solve certain PDEs.Definition 3.5. A convex lower semi-continuous Lagrangian L : V ×V ∗ →R ∪ {+∞} on a reflexive Banach space V is called self-dual on V × V ∗ ifL∗(p, u) = L(u, p),and we denote this class by L sd(V ).Remark 3.2. By definition of the Legendre transform of a Lagrangian L, itis clear that L(u, p) + L∗(p, u) ≥ 2〈u, p〉, and for a Lagrangian L ∈ L sd(V )we then haveL(u, p) ≥ 〈u, p〉, ∀ (u, p) ∈ V × V ∗. (3.1)However, the converse is not true, i.e. if L satisfies property (3.1) it doesnot necessarily imply that L is self-dual.Definition 3.6. The Lagrangian L satisfying (3.1) is called Fenchelian.Additionally, if L (non-self-dual) satisfiesL∗(p, u) ≥ L(u, p) ≥ 〈u, p〉, ∀ (u, p) ∈ V × V ∗,it is called a subself-dual Lagrangian.Example 1. The basic self-dual Lagrangians:253.3. Self-dual vector fields• A simple but important example of a self-dual Lagrangian on V × V ∗is defined via the formulaL(u, p) = ϕ(u) + ϕ∗(p)where ϕ is any convex l.s.c function on V .• Let Γ : V → V ∗ be a bounded linear operator which is skew-adjoint,and ϕ a convex l.s.c function, then L(u, p) = ϕ(u) + ϕ∗(Γu + p) isself-dual.More elaborate examples will be devised later, though all constructionswill be based on these important elements of L sd(V ).Operations on LagrangiansLet L and N be two given Lagrangians, we consider the following oper-ations on V × V ∗• Addition: (L⊕N)(u, p) = infr∈V ∗{L(u, r) +N(u, p− r)}.• Convolution: (L ? N)(u, p) = infz∈V{L(z, p) +N(u− z, p)}.Then we have the following lemma.Lemma 3.1. ([29] Proposition 3.4) Let V be a reflexive Banach space, andL and N two Lagrangians, then1. If Dom1(L) − Dom1(N) (resp,. Dom2(L∗) − Dom2(N∗)) contains aneighborhood of the origin, then(L⊕N)∗ = L∗ ? N∗, (resp., (L ? N)∗ = L∗ ⊕N∗)2. If L,N ∈ L sd(V ) such that Dom1(L) − Dom1(N) contains a neigh-borhood of the origin, then L⊕N and L ? N are also in L sd(V ).3.3 Self-dual vector fieldsIn this section, we introduce the concept of self-dual vector fields, whichare the natural extension of subdifferential of convex lower semi-continuousfunctions. Moreover, it is proved that self-dual Lagrangians play the role ofpotentials of maximal monotone vector fields in a way similar to how convexenergies are the potentials of their own subdifferentials.263.3. Self-dual vector fieldsDefinition 3.7. Let L : V × V ∗ → R ∪ {+∞} be a Lagrangian, its sym-metrized vector field at u ∈ V is defined as the -possibly empty- subset of V ∗given by∂¯L(u) = {p ∈ V ∗; L(u, p) + L∗(p, u) = 2〈u, p〉}If L is convex and l.s.c. on V × V ∗, then∂¯L(u) = {p ∈ V ∗; (p, u) ∈ ∂L(u, p)}If now L is a self-dual Lagrangian, then∂¯L(u) = {p ∈ V ∗; L(u, p)− 〈u, p〉 = 0}A self-dual vector field is any map F : Dom(F ) ⊂ V → 2V ∗ if there existsa self-dual Lagrangian L on V × V ∗ such that F (u) = ∂¯L(u) for everyu ∈ Dom(F ). A key point is that self-dual Lagrangians on phase spacenecessarily satisfy L(u, p) ≥ 〈u, p〉 for all (u, p) ∈ V × V ∗, and therefore thezeros of a self-dual vector field (i.e, 0 ∈ ∂¯L(u)) can be obtained by simplyminimizing the functional I(u) = L(u, 0) and proving that the value of theminimum is actually zero.For a general Lagrangian L we also define the vector fieldδL(u) = {p ∈ V ∗; L(u, p)− 〈u, p〉 = 0}.It is easy to show that if L is Fenchelian i.e. it satisfies L(u, p) ≥ 〈u, p〉,for every (u, p) ∈ V × V ∗, then δL(u) ⊂ ∂¯L(u); and clearly, if L is self-dualthen δL = ∂¯L.Examples• For a basic self-dual Lagrangian of the form L(u, p) = ϕ(u) + ϕ∗(p),where ϕ is a convex l.s.c function on V , and ϕ∗ is its Legendre conju-gate on V ∗, we have∂¯L(u) = ∂ϕ(u).In order to solve equations of the form 0 ∈ ∂ϕ(u), the correspond-ing variational problem reduces to minimizing the convex functionalI(u) = L(u, 0) = ϕ(u) + ϕ∗(0).• More interesting examples of self-dual Lagrangians are of the formL(u, p) = ϕ(u) + ϕ∗(−Γu+ p), where ϕ is a convex and l.s.c functionon V and Γ : V → V ∗ is a skew-adjoint operator. The correspondingself-dual vector field is then∂¯L(u) = Γu+ ∂ϕ(u).273.3. Self-dual vector fieldsProposition 3.3. For any convex lower semi-continuous Lagrangian L onV × V ∗, the map u 7→ ∂¯L(u) is monotone.Proposition 3.4. Let L be a self-dual Lagrangian on V × V ∗, then thefollowing are equivalent:1. p ∈ ∂¯L(u)2. The infimum of the functional Ip(u) = L(u, p)− 〈u, p〉 over all u ∈ Vis zero and is attained at some v ∈ V .The following lemmas are used to prove how to associate a self-dualLagrangian to a maximal monotone operator in such a way that the oper-ator coincides with the vector field corresponding to the Lagrangian. TheLagrangian LA defined in (3.2) is called the Fitzpatrick function.Lemma 3.2. Let A : Dom(A) ⊂ V → 2V ∗ be a maximal monotone operator.Consider on V × V ∗ the Lagrangian LA defined byLA(u, p) = sup{〈p, v〉+ 〈q, u− v〉; (v, q) ∈ G(A)}. (3.2)Then1. if Dom(A) 6= ∅, LA is convex and lower semi-continuous on V × V ∗.2. LA = H∗A where HA(u, p) ={〈u, p〉 if (u, p) ∈ G(A)+∞ elsewhere.3. for every u ∈ Dom(A), we have A = ∂¯LA = δLA. Moreover, LA issubself-dual i.e.L∗A(p, u) ≥ LA(u, p) ≥ 〈u, p〉, for every (u, p) ∈ V × V ∗.The following lemma is on interpolating convex functions, which is some-times in the literature called the proximal average of two convex functions.Lemma 3.3. Let f1, f2 : V → R ∪ {+∞} be two convex l.s.c. functions ona reflexive Banach space V . For u ∈ V , consider the proximal average of f1and f2 as the followingf¯(u) := inf{12f1(u1) +12f2(u2) +18‖u1− u2‖2; u1, u2 ∈ V, u = 12(u1 + u2)}.Then the following properties hold283.3. Self-dual vector fields1. (12f∗1 +12f∗2 )∗ ≤ f¯ ≤ 12f1 + 12f2, which implies that if f1 ≤ f2 thenf1 ≤ f¯ ≤ f2.2. Legendre dual. For p ∈ V ∗, we have(f¯)∗(p) := inf{12f∗1 (p1)+12f∗2 (p2)+18‖p1−p2‖2; p1, p2 ∈ V ∗, p = 12(p1+p2)}.3. Denote the proximal average between f1 and f2 by f¯ =: [f1, f2], then[f1, f2]∗ = [f∗1 , f∗2 ].We are now ready to state the theorem which establishes a one-to-onecorrespondence between maximal monotone operators and self-dual vectorfields.Theorem 3.2. (Self-dual vector fields and maximal monotone op-erators) If A : D(A) ⊂ V → 2V ∗ is a maximal monotone operator with anon-empty domain, then there exists a self-dual Lagrangian N on V × V ∗such that A = ∂¯N . Conversely, if N is a proper self-dual Lagrangian on areflexive Banach space V × V ∗, then the vector field u 7→ ∂¯N(u) is maximalmonotone.Proof. Since A is maximal monotone, we first via Lemma 3.2 associate thesubself-dual Lagrangian LA so thatA = ∂¯LA and L∗A(p, u) ≥ LA(u, p) ≥ 〈u, p〉. (3.3)Now we claim that the desired self-dual Lagrangian corresponding to the op-erator A is the proximal average between LA and L∗A, namely N := [LA, L∗A]given byN(u, p) = inf{12LA(u1, p1) +12L∗A(p2, u2)+18‖u1 − u2‖2 + 18‖p1 − p2‖2; (u, p) = 12(u1, p1) + (u2, p2)}.By Lemma 3.3, since LA ≤ L∗A, we haveL∗A(p, u) ≥ N(u, p) ≥ LA(u, p) for every (u, p) ∈ V × V ∗,also it is clear that N is self-dual sinceN∗ = [LA, L∗A]∗ = [L∗A, L∗∗A ] = [LA, L∗A] = N.293.4. Variational principle for self-dual functionalsNow we prove ∂¯N = ∂¯LA. First, take p ∈ ∂¯N(u), since N is self-dual,this implies that N(u, p) − 〈u, p〉 = 0. Now from LA(u, p) ≤ N(u, p) wededuce that LA(u, p) ≤ 〈u, p〉 which by (3.3) implies LA(u, p) = 〈u, p〉, thusp ∈ δLA(u) ⊂ ∂¯LA.Now if p ∈ ∂¯LA(u), then we have LA(u, p) + L∗A(p, u) = 2〈u, p〉, also byLemma 3.2 we have (u, p) ∈ G(A). Monotonicity of A implies that〈u, p〉 ≥ 〈v, p〉+ 〈u− v, q〉,which from definition of LA yields LA(u, p) ≤ 〈u, p〉 and with (3.3) we con-clude LA(u, p) = 〈u, p〉 = HA(u, p), thus L∗A(p, u) = 〈u, p〉 and thereforeN(u, p) = 〈u, p〉.Hence p ∈ ∂¯N(u). By Lemma 3.2 since A = ∂¯LA then Au = ∂¯N(u) forevery (u, p) ∈ V × V ∗ with u ∈ Dom(A), and this completes the proof.For the proof of the converse we refer the reader to [30].3.4 Variational principle for self-dual functionalsSeveral differential systems often because of lack of self-adjointness or lin-earity cannot be expressed as Euler-Lagrange equations, but they can bewritten in the form0 ∈ ∂¯L(u),where L is a self-dual Lagrangian on phase space V ×V ∗ and V is a reflexiveBanach space. These are the completely self-dual systems, and a solutionto these systems can be considered as minimizers of a completely self-dualfunctional I for which the minimum value is 0.The fact that the infimum of the functional I is zero follows from thebasic duality theory in convex analysis, which in the self-dual setting leadsto a situation where the value of the dual problem is exactly the negativeof the value of the primal problem, hence leading to zero as soon as there isno duality gap.3.4.1 Evolution triples and self-dual LagrangiansA common framework for PDEs and evolution equations is the so-calledevolution triple of Gelfand, which is a natural setting to obtain more regularsolutions that are valued in suitable Sobolev spaces, as opposed to just303.4. Variational principle for self-dual functionalsL2; moreover to relax the restrictive coercivity condition on the underlyingHilbert space. It consists of a Hilbert space H with 〈 , 〉H as its scalarproduct, and the space V which is equipped with the norm ‖·‖V that makesit a reflexive Banach space and such that the canonical injection V → H iscontinuous. We identify the Hilbert space H with its dual H∗ and we injectH in V ∗ via the following:Let 〈 , 〉∗ denotes the dualization between V ∗ and V . For each h ∈ H,the map Th : u ∈ V → 〈h, u〉H is a continuous linear functional on V insuch a way that〈Th, u〉∗ = 〈h, u〉H ∀h ∈ H, u ∈ V.One can easily see that T : H → V ∗ is continuous, one-to-one, and thatT (H) is dense in V ∗. In other words, one can place H in V ∗ such thatV ⊂ H ∼= H∗ ⊂ V ∗, where the injections are continuous and with denserange. We note that with such an identification the duality 〈f, u〉∗ coincideswith the scalar product 〈f, u〉H as soon as f ∈ H and u ∈ V .A typical evolution triple is V := H10 (D) ⊂ H := L2(D) ⊂ V ∗ :=H−1(D), where D is a bounded domain in Rn. The following lemma explainsthe connection between self-duality on H and V .Lemma 3.4. ([29] Lemma 3.4) Let V ⊂ H ⊂ V ∗ be an evolution triple, andsuppose L : V × V ∗ → R ∪ {+∞} is a self-dual Lagrangian on the Banachspace V , that satisfies for some C1, C2 > 0 and r1 ≥ r2 > 1,C2(‖u‖r2V − 1) ≤ L(u, 0) ≤ C1(1 + ‖u‖r1V ) for all u ∈ V.Then, the Lagrangian defined on H ×H byL¯(u, p) :={L(u, p) u ∈ V+∞ u ∈ H\Vis self-dual on the Hilbert space H ×H.3.4.2 Primal and dual convex optimization problemConsider the primal problem to be the minimization of a convex lower semi-continuous function I that is bounded below on a Banach space V , i.e.infu∈VI(u), (3.4)313.4. Variational principle for self-dual functionalswe consider the convex lower semi-continuous Lagrangian L : V × Y →R ∪ {+∞} such that I can be written asI(u) = L(u, 0) for all u ∈ V.For any p ∈ Y , we consider the perturbed minimization problem(Pp) infu∈VL(u, p), (3.5)for which (P0) is clearly the initial primal problem (3.4). Now we considerthe dual problem on V ∗ × Y ∗(P∗) supp∗∈Y ∗−L∗(0, p∗), (3.6)and we define the function h : Y → R∪{+∞} on the space of perturbationsYh(p) = infu∈VL(u, p) for all p ∈ Y.The following proposition summarizes the relationship between the pri-mal and dual problems and the behavior of the value function h. A proofcan be found in [29].Proposition 3.5. Assume L is a proper convex lower semi-continuous La-grangian that is bounded below on V × Y . Then, the following assertionshold:1. −∞ < supp∗∈Y ∗− L∗(0, p∗) ≤ infu∈VL(u, 0) < +∞.2. h is a convex function on Y such thath∗(p∗) = L∗(0, p∗) for every p∗ ∈ Y ∗,and h∗∗(0) = supp∗∈Y ∗ −L∗(0, p∗).3. h is lower semi-continuous at 0 (i.e. the problem (3.4) is normal) ifand only if there is no duality gap, i.e. ifsupp∗∈Y ∗−L∗(0, p∗) = infu∈VL(u, 0).4. h is subdifferentiable at 0 if and only if (P0) is normal and (P∗) hasat least one solution. Moreover, the set of solutions for (P∗) is equalto ∂h∗∗(0).323.4. Variational principle for self-dual functionals5. If for some u0 ∈ V the function p → L(u0, p) is bounded on a ballcentered at 0 in Y , then (P0) and (P∗) has at least one solution.Before stating the minimization principle of completely self-dual func-tionals, we should consider the following relaxed version of self-duality whichis sufficient for our purpose in this setting.Definition 3.8. Let L be a Lagrangian on V ×V ∗, we say that L is partiallyself-dual ifL∗(0, u) = L(u, 0).Definition 3.9. A function I : V → R ∪ {∞} is said to be a completelyself-dual functional on the Banach space V if there exists a partially self-dualLagrangian L on V × V ∗ such thatI(u) = L(u, 0), for u ∈ V.The following theorem is the key variational principle for minimizationof self-dual Lagrangians on V ×V ∗, and in particular it proves that the valueof the minimum is 0.Theorem 3.3. Let L be partially self-dual (convex lower semi-continuous)on V ×V ∗, and also assume that the mapping u→ L(u, 0) is coercive in thesense thatlim‖u‖→∞L(u, 0)‖u‖ = +∞ (3.7)then there exists u¯ ∈ V such thatI(u¯) = infu∈VL(u, 0) = 0Proof. Following the duality theory in convex optimization described inProposition 3.5, we consider the minimization problem h(p) = infu∈VL(u, p) insuch a way that h(0) = infu∈VL(u, 0) is the initial problem, and then the dualproblem is supv∈V− L∗(0, v).By assertion (1) in Proposition 3.5 and noting that L is partially self-dualwe haveinfu∈VL(u, 0) ≥ supv∈V−L∗(0, v) = supv∈V−L(v, 0) = − infv∈VL(v, 0),thusinfu∈VL(u, 0) ≥ 0.333.5. The class of antisymmetric HamiltoniansAlso by assertion (2) of Proposition 3.5, h is convex on V ∗ and its Legen-dre transform on V satisfies h∗(v) = L∗(0, v) = L(v, 0), which by coercivitycondition (3.7) implies that h∗ is coercive, and this means that h is boundedabove on neighborhoods of 0 in V ∗. Therefore, h is subdifferentiable at 0,and there exists u¯ ∈ ∂h(0), which is equivalent to h(0) + h∗(u¯) = 0.Now we haveinfu∈VL(u, 0) = h(0) = −h∗(u¯) = −L∗(0, u¯) = −L(u¯, 0) ≤ − infu∈VL(u, 0),which implies infu∈VL(u, 0) ≤ 0.Hence, we conclude that infu∈VL(u, 0) = L∗(0, u¯) =0, and (3.4) is attained at u¯ ∈ V , i.e.I(u¯) = L(u¯, 0) = infu∈VL(u, 0) = 0.3.5 The class of antisymmetric HamiltoniansThere are several examples of PDEs which cannot be expressed in terms ofcompletely self-dual systems but they can be written in the form0 ∈ Λu+ ∂¯L(u),where L is a self-dual Lagrangina on V × V ∗, and Λ : D(Λ) ⊂ V → V ∗ isa, not necessarily linear, operator. They can be solved by minimizing thefunctionalI(u) = L(u,−Λu) + 〈Λu, u〉,on V by showing that their infimum is zero and that it is attained. In[29], these are called self-dual functionals, which take the form I(u) =supv∈V M(u, v) where M is an antisymmetric Hamiltonian on V × V . Inthe following we first introduce the Hamiltonian corresponding to a self-dualLagrangian, and in turn the class of antisymmetric Hamiltonians and theirvariational framework.For each Lagrangian L on V ×V ∗, we can define its corresponding Hamil-tonian HL : V × V → R ∪ {+∞} byHL(u, v) = supp∈V ∗{〈v, p〉 − L(u, p)},343.5. The class of antisymmetric Hamiltonianswhich is the Legendre transform in the second variable and its effectivedomain isDom1(HL) : = {u ∈ V ;HL(u, v) > −∞, for some v ∈ V }= {u ∈ V ;L(u, p) < +∞, for some p ∈ V ∗}= Dom1(L).It is easy to see that if L is a self-dual Lagrangian on V × V ∗, then itsHamiltonian on V × V satisfies the following properties:• HL is concave in u and convex lower semi-continuous in v.• HL(v, u) ≤ −HL(u, v) for all u, v ∈ V.Definition 3.10. Let G be a convex subset of a reflexive Banach space V.A functional M : G×G→ R ∪ {+∞} is said to be an antisymmetric (AS)Hamiltonian on G×G if it satisfies the following conditions:1. For every u ∈ G, the function v →M(u, v) is proper concave.2. For every v ∈ G , the function u→M(u, v) is weakly lower semicon-tinuous.3. M(u, u) ≤ 0 for every u ∈ V .3.5.1 Variational principle for self-dual functionalsDefinition 3.11. (Self-dual functional) Let I : V → R ∪ {+∞} be a func-tional on a Banach space V .1. I is self-dual on a convex set G ⊂ V if it is non-negative and if thereexists an AS-Hamiltonian M : G×G→ R ∪ {+∞} such thatI(u) = supv∈GM(u, v), for every u ∈ G.2. A self-dual functional I is strongly coercive on G if for some v0 ∈ Gthe set G0 = {u ∈ G; M(u, v0) ≤ 0} is bounded in V , where M is thecorresponding AS-Hamiltonian in part (1).353.5. The class of antisymmetric HamiltoniansExample 2. For a self-dual Lagrangian L, the completely self-dual func-tional I(u) = L(u, 0) as in the preceding section is self-dual. In fact;I(u) = L(u, 0) = sup{HL(v, u); v ∈ Dom1(L)} for every u ∈ Dom1(L),and the strong corecivity is equivalent to the condition that, for some v0 ∈ Vwe havelim‖u‖→+∞HL(v0, u) = +∞.In particular, if L(u, p) = ϕ(u) + ϕ∗(p) for ϕ a proper, convex and lowersemicontinuous function, the strong coercivity is simply equivalent to thecoercivity of ϕ.The following theorem states the main variational principle for self-dualfunctionals.Theorem 3.4. If I : G → R ∪ {+∞} is a self-dual functional that isstrongly coercive on a closed convex subset G of a reflexive Banach space V ,then there exists u¯ ∈ G such thatI(u¯) = infu∈GI(u) = 0.The proof is based on the important min-max principle that is due toK. Fan [22], and it builds upon the following primary lemma.Lemma 3.5. Let G be a closed convex subset of a reflexive Banach spaceV , and consider M : G×G→ R ∪ {+∞} to be a functional such that(i) For each v ∈ G, the map u→M(u, v) is weakly l.s.c on G.(ii) For each u ∈ G, the map v →M(u, v) is concave on G.(iii) There exists γ ∈ R, such that M(u, u) ≤ γ for every u ∈ G.(iv) There exists v0 ∈ G, such that G0 = {u ∈ G : M(u, v0) ≤ γ} isbounded.Then there exists u¯ ∈ G, such that M(u¯, v) ≤ γ for all v ∈ G.Proof of Theorem 3.4. I is a self-dual functional which means that it isnon-negative and for every u ∈ G, it can be written asI(u) = supv∈GM(u, v),363.5. The class of antisymmetric Hamiltonianswhere M is an AS-Hamiltonian on G × G. By the definition of an AS-Hamiltonian, M satisfies conditions (i)-(iii) of Lemma 3.5 with γ = 0. con-dition (iv) is also satisfied with the strong coercivity assumption. Therefore,Lemma 3.5 yields the existence of u¯ ∈ G such that M(u¯, v) ≤ 0 for all v ∈ Gand hence0 ≥ supv∈GM(u¯, v) = I(u¯) ≥ 0.As established in [28], the Hamiltonian formulation allows for the min-imization of direct sums of self-dual functionals. The following variationalprinciple is useful in the case when non-linear and unbounded operators areinvolved.Theorem 3.5. (Ghoussoub [29]) Consider three reflexive Banach spacesZ,X1, X2 and operators A1 : D(A1) ⊂ Z → X1 , Γ1 : D(Γ1) ⊂ Z → X∗1 ,A2 : D(A2) ⊂ Z → X2, and Γ2 : D(Γ2) ⊂ Z → X∗2 , such that A1 andA2 are linear, while Γ1 and Γ2 –not necessarily linear– are weak-to-weakcontinuous. Suppose G is a closed linear subspace of Z such that G ⊂D(A1)∩D(A2)∩D(Γ1)∩D(Γ2), while the following properties are satisfied:1. The image of G0 := Ker(A2) ∩G by A1 is dense in X1.2. The image of G by A2 is dense in X2.3. u→ 〈A1u,Γ1u〉+ 〈A2u,Γ2u〉 is weakly upper semi-continuous on G.Let Li, i = 1, 2 be self-dual Lagrangians on Xi ×X∗i such that the Hamilto-nians HLi are continuous in the first variable on Xi.(i) The functionalI(u) = L1(A1u,Γ1u)− 〈A1u,Γ1u〉+ L2(A2u,Γ2u)− 〈A2u,Γ2u〉is self-dual on G, and its corresponding AS-Hamiltonian on G×G isM(u, z) = HL1(A1z,A1u) + 〈A1(z − u),Γ1u〉+HL2(A2z,A2u) + 〈A2(z − u),Γ2u〉.(ii) Consequently under the coercivity condition,lim‖u‖→∞u∈GHL1(0, A1u)− 〈A1u,Γ1u〉+HL2(0, A2u)− 〈A2u,Γ2u〉 = +∞,(3.8)373.5. The class of antisymmetric HamiltoniansI attains its minimum at a point v ∈ G such that I(v) = 0, andΓ1(v) ∈ ∂¯L1(A1v),Γ2(v) ∈ ∂¯L2(A2v). (3.9)Proof. (i) First, we show that M(u, z) is actually an AS-Hamiltonian. Tothis end, we prove that M satisfies the properties given in Definition 3.10.For each u ∈ G, we have that z → M(u, z) is concave since Ai’s are lin-ear, also z → HLi(Aiz,Aiu) is concave. For each z ∈ G, the functionu → M(u, z) is weakly lower semi-continuous due to upper semi-continuityof u → ∑2i=1〈Aiu,Γiu〉, and the fact that Γi’s are weak-to-weak continu-ous and u → HLi(Aiz,Aiu) is weakly lower semi-continuous for linear Ai.Finally, since HLi(x, y) ≤ −HLi(y, x) for all x, y ∈ Xi, we deduce thatM(u, u) ≤ 0.Next, we note that for every u ∈ D(Ai) ∩D(Γi) ⊂ Z we haveLi(Aiu,Γiu) = L∗i (Γiu,Aiu)= supr∈Xisupp∈X∗i{〈r,Γiu〉+ 〈p,Aiu〉 − Li(r, p)}= supr∈Xi{〈r,Γiu〉+ supp∈X∗i{〈p,Aiu〉 − Li(r, p)}}= supr∈Xi{〈r,Γiu〉+HLi(r,Aiu)}.(3.10)Since Ai are liner operators, taking into account condition (1), we writesupz∈GM(u, z) = supz∈G{〈A1z,Γ1u〉+HL1(A1z,A1u) + 〈A2z,Γ2u〉+HL2(A2z,A2u)}− 〈A1u,Γ1u〉 − 〈A2u,Γ2u〉= supz∈G,z0∈G0{〈A1z,Γ1u〉+HL1(A1z,A1u)+ 〈A2(z + z0),Γ2u〉+HL2(A2(z + z0), A2u)}− 〈A1u,Γ1u〉 − 〈A2u,Γ2u〉= supw∈G,z0∈G0{〈A1(w − z0),Γ1u〉+HL1(A1(w − z0), A1u)+ 〈A2w,Γ2u〉+HL2(A2w,A2u)}− 〈A1u,Γ1u〉 − 〈A2u,Γ2u〉383.5. The class of antisymmetric Hamiltonians= supw∈G,r∈X1{〈A1w + r,Γ1u〉+HL1(A1w + r,A1u)+ 〈A2w,Γ2u〉+HL2(A2w,A2u)}− 〈A1u,Γ1u〉 − 〈A2u,Γ2u〉= supw∈G,x∈X1{〈x,Γ1u〉+HL1(x,A1u) + 〈A2w,Γ2u〉+HL2(A2w,A2u)}− 〈A1u,Γ1u〉 − 〈A2u,Γ2u〉In view of condition (2) and (3.10) we then havesupz∈GM(u, z) = supx∈X1supy∈X2{〈x,Γ1u〉+HL1(x,A1u) + 〈y,Γ2u〉+HL2(y,A2u)}− 〈A1u,Γ1u〉 − 〈A2u,Γ2u〉= supx∈X1{〈x,Γ1u〉+HL1(x,A1u)}− 〈A1u,Γ1u〉+ supy∈X2{〈y,Γ2u〉+HL2(y,A2u)}− 〈A2u,Γ2u〉= L1(A1u,Γ1u)− 〈A1u,Γ1u〉+ L2(A2u,Γ2u)− 〈A2u,Γ2u〉= I(u).For part (ii), it suffices to apply Theorem 3.4 to get that I(v) = 0 forsome v ∈ G. Now use the fact that for self-dual Li, i = 1, 2, we haveLi(Aiu,Γiu)− 〈Aiu,Γiu〉 ≥ 0 to conclude (3.9).39Chapter 4Self-dual variational principlefor stochastic partialdifferential equations withadditive noise4.1 IntroductionIn Chapter 3, we provided the reader with the variational framework corre-sponding to self-dual Lagrangians and we observed that the minimization of(completely) self-dual functionals leads to existence of solutions to certainPDEs. A main application of this approach is to solve equations that arenot in the standard Euler-Lagrange form, namely the equations for whichthe classical variational setting does not apply. One important class ofsuch equations is the family of stochastic partial differential equations ona probability space (Ω,F ,P) equipped with normal filtration (Ft)t. Theevolutionary nature of the stochastic equations together with the presenceof the stochastic integral with respect to a Wiener process make them ofnon-variational structure.To introduce the method, as stated in Chapter 1, we first consider thesimple case {du(t) = −∂ϕ(t, u(t))dt+B(t)dW (t)u(0) = u0.(4.1)where ϕ : [0, T ]×H → R∪{+∞} is a random and progressively measurablefunction such that for every t ∈ [0, T ], the function ϕ(t, ·) is convex and lowersemi-continuous on a Hilbert space H, and the stochastics is driven by agiven progressively measurable additive noise coefficient B : Ω× [0, T ]→ H.The main idea is that a solution for (4.1) is in fact the minimum of the404.1. Introductionfollowing functional on A2H ,I(u) = E{∫ T0Lϕ(u(t),−u˜(t)) dt+ 12∫ T0MB(Fu(t),−Fu(t)) dt+ `u0(u(0), u(T ))},whereA2H ={u : ΩT → H; u(t) = u0 +∫ t0u˜(s)ds+∫ t0Fu(s)dW (s)},is the Itoˆ space over H, for progressively measurable processes u˜ and Fu,and Lϕ, MB and `u0 are the Lagrangians given by (1.3), (1.4) and (1.5)respectively. We can then apply the self-dual variational principle given inTheorem 3.3 to prove that I attains its infimum on v ∈ A2H , and moreover,the value of the infimum is 0 to obtain0 = I(v) = E∫ T0(ϕ(t, v) + ϕ∗(t,−v˜(t)))dt+ E(12‖v(0)‖2H +12‖v(T )‖2H − 2〈u0, v(0)〉+ ‖u0‖2H)+ E∫ T0(12‖Fv(t)− 2B(t)‖2H +12‖Fv(t)‖2H − 2〈Fv(t), B(t)〉)dt,By using Itoˆ’s formula and adding and subtracting the term E∫ T0 〈v(t), v˜(t)〉dt,we can rewrite I(v) in the form0 = I(v) = E∫ T0(ϕ(t, v(t)) + ϕ∗(t,−v˜(t)) + 〈v(t), v˜(t)〉)dt+ 2E∫ T0‖Fv −B‖2H dt+ E ‖v(0)− u0‖2H ,and the fact that the first integrand is non-negative yields that for almostall t ∈ [0, T ], P-a.s.ϕ(t, v(t)) + ϕ∗(t,−v˜(t)) + 〈v(t), v˜(t)〉 = 0, hence −v˜(t) ∈ ∂ϕ(v(t)),and the two other identities readily give that B = Fv and v(0) = u0. Thisimplies that v(t) = u0 −∫ t0 ∂ϕ(s, v(s))ds +∫ t0 B(s)dW (s) is a solution to(4.1).The self-dual variational calculus allows to apply the above approachin much more generality. The special Lagrangians Lϕ, `u0 and M can be414.2. Lifting random self-dual Lagrangians to Itoˆ path spacesreplaced by much more general self-dual Lagrangians. In fact, we show thatdue to the correspondence between the class of self-dual Lagrangians andmaximal monotone operators, one can consider general equations of the form{du(t) = −A(t, u(t))dt+B(t)dW (t)u(0) = u0,(4.2)where u0 ∈ L2(Ω,F0,P;H) for the Hilbert space H, W (t) is a real-valuedWiener process on a complete probability space (Ω,F ,P) with normal fil-tration (Ft)t, and where B : [0, T ]×Ω→ H is a given Hilbert-space valuedprogressively measurable process. A : Ω × [0, T ] × V → 2V ∗ is a time-dependent progressively measurable –possibly set-valued– maximal mono-tone map, where V is a Banach space such that V ⊂ H ⊂ V ∗ forms aGelfand evolution triple.In Section 4.2, we show how one can lift self-dual Lagrangians from statespace to Lp-spaces and then to Itoˆ spaces of stochastic processes. In Section4.3, we give a variational resolution for Equation (4.2) by using the basicminimization principle (Theorem 3.3) for self-dual Lagrangians. Section 4.4contains applications to classical SPDEs such as the following stochasticevolution driven by a diffusion and a transport operator,{du = (∆u+ a(x) · ∇u)dt+B(t)dW on [0, T ]×Du(0) = u0 on D,(4.3)where a : D → Rn is a smooth vector field with compact support in D,such that div(a) ≥ 0. Other examples include the stochastic porous mediaequation, but also quasi-linear equations involving the p-Laplacian (2 ≤ p <+∞), that is{du = (∆pu− u|u|p−2)dt+B(t)dW on D × [0, T ]u(0) = u0 on ∂D.(4.4)4.2 Lifting random self-dual Lagrangians to Itoˆpath spacesLet V be a reflexive Banach space, and T ∈ [0,∞) be fixed. Consider acomplete probability space (Ω,F ,P) with a normal filtration Ft, t ∈ [0, T ],and let Lα(Ω× [0, T ];V ) be the space of Bochner integrable functions fromΩT := Ω× [0, T ] into V with the norm ‖u‖αLαV := E∫ T0 ‖u(t)‖αV dt. We mayuse the shorter notation LαV (ΩT ) := Lα(Ω× [0, T ];V ) in the sequel.424.2. Lifting random self-dual Lagrangians to Itoˆ path spacesDefinition 4.1. A self-dual ΩT -dependent convex Lagrangian on V ×V ∗ isa function L : ΩT × V × V ∗ → R ∪ {+∞} such that:1. L is progressively measurable with respect to the σ-field generated bythe products of Ft and Borel sets in [0, t] and V × V ∗, i.e. for everyt ∈ [0, T ], L(t, ·, ·) is Ft ⊗ B([0, t])⊗ B(V )⊗ B(V ∗)-measurable.2. For each t ∈ [0, T ], P-a.s. the function L(t, ·, ·) is convex and lowersemi-continuous on V × V ∗.3. For any t ∈ [0, T ], we have P-a.s. L∗(t, p, u) = L(t, u, p) for all(u, p) ∈ V × V ∗, where L∗ is the Legendre transform of L in the lasttwo variables.To each ΩT -dependent Lagrangian L on ΩT ×V ×V ∗, one can associate thecorresponding Lagrangian L on the path space LαV (ΩT ) × LβV ∗(ΩT ), where1α +1β = 1, to beL(u, p) := E∫ T0L(t, u(t), p(t)) dt,and with the duality between LαV (ΩT ) and LβV ∗(ΩT ) given by〈u, p〉 = E∫ T0〈u(t), p(t)〉V,V ∗dt.The associated Hamiltonian on LαV (ΩT )× LαV (ΩT ) will then beHL(u, v) = sup{E∫ T0{〈v(t), p(t)〉 − L(t, u(t), p(t))}dt ; p ∈ LβV ∗(ΩT )}.The Legendre dual of a ”lifted” Lagrangian in both variables naturally liftsto the space of paths LαV (ΩT )× LβV ∗(ΩT ) viaL∗(q, v) = supu∈LαV (ΩT )p∈LβV ∗ (ΩT ){E∫ T0{〈q(t), u(t)〉+ 〈v(t), p(t)〉 − L(t, u(t), p(t))} dt}.The following proposition is standard. See for example [20].Proposition 4.1. Suppose that L is an ΩT -dependent Lagrangian on V ×V ∗, and L is the corresponding Lagrangian on the path space LαV (ΩT ) ×LβV ∗(ΩT ). Then,1. L∗(p, u) = E ∫ T0 L∗(t, p(t), u(t))dt.2. HL(u, v) = E∫ T0 HL(t, u(t), v(t))dt.434.2. Lifting random self-dual Lagrangians to Itoˆ path spaces4.2.1 Self-dual Lagrangians associated to progressivelymeasurable monotone fieldsConsider now a progressively measurable –possibly set-valued– maximalmonotone map that is a map A : ΩT × V → 2V ∗ that is measurable foreach t, with respect to the product σ-field Ft ⊗ B([0, t]) ⊗ B(V ), and suchthat for each t ∈ [0, T ], P-a.s., the vector field Aω,t := A(t, ω, ·) is maxi-mal monotone on V . By Theorem 3.2, one can associate to the maximalmonotone maps Aω,t, self-dual Lagrangians LAω,t on V × V ∗, in such a waythatAω,t = ∂¯LAω,t for every t ∈ [0, T ], and P-a.s.This correspondence can be done measurably in such a way that if A isprogressively measurable, then the same holds for the corresponding ΩT -dependent Lagrangian L. We can then lift the random Lagrangian to thespace LαV (ΩT )× LβV ∗(ΩT ) viaLA(u, p) = E∫ T0LAω,t(u(ω, t), p(ω, t))dt.Boundedness and coercivity conditions on A translate into correspondingconditions on the representing Lagrangians as follows. For simplicity, weshall assume throughout that the monotone operators are single-valued,though the results apply for general vector fields.Lemma 4.1. ([32]) Let Aω,t be the maximal monotone operator as abovewith the corresponding potential Lagrangian LAω,t . Assume that for allu ∈ V, dt⊗ P a.s., Aω,t satisfies〈Aω,tu, u〉 ≥ max{c1(ω, t)‖u‖αV −m1(ω, t), c2(ω, t)‖Aω,tu‖βV ∗ −m2(ω, t)},(4.5)where c1, c2 ∈ L∞(ΩT , dt ⊗ P) and m1,m2 ∈ L1(ΩT , dt ⊗ P). Then thecorresponding Lagrangians satisfy the following:C1(ω, t)(‖u‖αV +‖p‖βV ∗−n1(ω, t) ≤ LAw,t(u, p) ≤ C2(ω, t)(‖u‖αV +‖p‖βV ∗ +n2(ω, t)),for some C1, C2 ∈ L∞(ΩT ) and n1, n2 ∈ L1(ΩT ).The lifted Lagrangian on the Lα-spaces then satisfy for some C1, C2 > 0,C1(‖u‖αLαV (ΩT )+‖p‖βLβV ∗ (ΩT )−1) ≤ LA(u, p) ≤ C2(1+‖u‖αLαV (ΩT )+‖p‖βLβV ∗ (ΩT )).444.2. Lifting random self-dual Lagrangians to Itoˆ path spaces4.2.2 Itoˆ path spaces over a Hilbert spaceIn view of the definition of a Hilbert-valued Wiener process given in Section2.5, we suppose that U = R and hence W is a real-valued Brownian motion.We now recall Itoˆ’s formula.Proposition 4.2. ([46], [48]) Let H be a Hilbert space with 〈 , 〉H as itsscalar product. Fix x0 ∈ L2(Ω,F0,P;H), and let y ∈ L2(ΩT ;H), Z ∈L2(ΩT ;H) be two progressively measurable processes. Define the H-valuedprocess u asu(t) := x0 +∫ t0y(s)ds+∫ t0Z(s)dW (s). (4.6)Then, the following holds:1. u is a continuous H-valued adapted process such thatE(supt∈[0,T ] ‖u(t)‖2H)<∞.2. (Itoˆ’s formula) For all t ∈ [0, T ],‖u(t)‖2H = ‖x0‖2H + 2∫ t0〈y(s), u(s)〉Hds+∫ t0‖Z(s)‖2Hds+ 2∫ t0〈u(s), Z(s)〉HdW (s),and consequentlyE(‖u(t)‖2H) = E(‖x0‖2H) + E∫ t0(2〈y(s), u(s)〉H + ‖Z(s)‖2H)ds.More generally, the following integration by parts formula holds. For twoprocesses u and v of the form:u(t) = u(0)+∫ t0u˜(s)ds+∫ t0Fu(s)dW (s), v(t) = v(0)+∫ t0v˜(s)ds+∫ t0Gv(s)dW (s),we haveE∫ T0〈u(t), v˜(t)〉dt =− E∫ T0〈v(t), u˜(t)〉dt− E∫ T0〈Fu(t), Gv(t)〉dt+ E〈u(T ), v(T )〉H − E〈u(0), v(0)〉H . (4.7)454.2. Lifting random self-dual Lagrangians to Itoˆ path spacesNow we define the Itoˆ space A2H consisting of all H-valued processes of thefollowing form:A2H ={u :ΩT → H; u(t) = u(0) +∫ t0u˜(s)ds+∫ t0Fu(s)dW (s),for u(0) ∈ L2(Ω,F0,P;H), u˜ ∈ L2(ΩT ;H), Fu ∈ L2(ΩT ;H)},(4.8)where u˜ and Fu are both progressively measurable. We equip A2H with thenorm‖u‖2A2H = E(‖u(0)‖2H +∫ T0‖u˜(t)‖2H dt+∫ T0‖Fu(t)‖2H dt),so that it becomes a Hilbert space. Indeed, the following correspondence(x0, y, Z) ∈ L2(Ω;H)× L2(ΩT ;H)× L2(ΩT ;H)7→ x0 +∫ t0y(s)ds+∫ t0Z(s)dW (s) ∈ A2H ,(4.9)u ∈ A2H 7→ (u(0), u˜, Fu) ∈ L2(Ω;H)× L2(ΩT ;H)× L2(ΩT ;H),induces an isometry, since for two processes u, v ∈ A2H , Itoˆ’s formula appliedto u− v ∈ A2H yields that‖u(t)− v(t)‖2H = ‖u(0)− v(0)‖2H + 2∫ t0〈u˜(s)− v˜(s), u(s)− v(s)〉Hds+∫ t0‖Fu(s)− Fv(s)‖2Hds+ 2∫ t0〈u(s)− v(s), Fu(s)− Fv(s)〉HdWs,which means that u = v if and only if u(0) = v(0), Fu = Fv and u˜ = v˜. Wetherefore can and shall identify the Itoˆ space A2H with the product spaceL2(Ω;H)× L2(ΩT ;H)× L2(ΩT ;H).The dual space (A2H)∗ can also be identified with L2(Ω;H) × L2(ΩT ;H) ×L2(ΩT ;H). In other words, each p ∈ (A2H)∗ can be represented by the tripletp = (p0, p1(t), P (t)) ∈ L2(Ω;H)× L2(ΩT ;H)× L2(ΩT ;H),in such a way that the duality can be written as:〈u, p〉A2H×(A2H)∗ = E{〈p0, u(0)〉+∫ T0〈p1(t), u˜(t)〉dt+ 12∫ T0〈P (t), Fu(t)〉dt}.(4.10)464.2. Lifting random self-dual Lagrangians to Itoˆ path spaces4.2.3 Self-dual Lagrangians on Itoˆ spaces of randomprocessesWe now prove the following.Theorem 4.1. Let (Ω,F ,Ft,P) be a complete probability space with normalfiltration, and let L and M be two ΩT -dependent self-dual Lagrangians onH ×H, Assume ` is an Ω-dependent function on H ×H, such that P-a.s.`(ω, a, b) = `∗(ω,−a, b), (a, b) ∈ H ×H. (4.11)The Lagrangian on A2H × (A2H)∗ defined byL(u, p) = E{∫ T0L(u(t)− p1(t),−u˜(t)) dt+ `(u(0)− p0, u(T ))+12∫ T0M(Fu(t)− P (t),−Fu(t)) dt},(4.12)is then partially self-dual. Actually, it is self-dual on the subset A2H ×D ofA2H × (A2H)∗, where D := ({0} × L2H(ΩT )× L2H(ΩT )).Proof. Take (q, v) ∈ (A2H)∗ ×A2H with q an element in the dual space iden-tified with the triple (0, q1(t), Q(t)), thenL∗(q, v) = supu∈A2Hp∈(A2H)∗{〈q, u〉+ 〈v, p〉 − L(u, p)}= supu∈A2Hsupp0∈L2H(Ω)p1∈L2H(ΩT ),P∈L2H(ΩT )E{〈p0, v(0)〉+∫ T0(〈q1(t), u˜(t)〉+ 〈p1(t), v˜(t)〉)dt+12∫ T0(〈Q(t), Fu(t)〉+ 〈P (t), Gv(t)〉)dt−∫ T0L(u(t)− p1(t),−u˜(t)) dt− `(u(0)− p0, u(T ))− 12∫ T0M(Fu(t)− P (t),−Fu(t)) dt}.Make the following substitutions:u(t)− p1(t) = y(t) ∈ L2H(ΩT )u(0)− p0 = a ∈ L2H(Ω)Fu(t)− P (t) = J(t) ∈ L2H(ΩT ),474.2. Lifting random self-dual Lagrangians to Itoˆ path spacesto obtainL∗(q, v) = supu∈A2Hsupa∈L2H(Ω)supy∈L2H(ΩT )supJ∈L2H(ΩT )E{〈u(0)− a, v(0)〉 − `(a, u(T ))+∫ T0(〈q1(t), u˜(t)〉+ 〈u(t)− y(t), v˜(t)〉 − L(y(t),−u˜(t)))dt+12∫ T0〈Q(t), Fu(t)〉+ 〈Fu(t)− J(t), Gv(t)〉 −M(J(t),−Fu(t)) dt}.Use Itoˆ’s formula (4.7) for the processes u and v in A2H to getL∗(q, v) = supu∈A2Hsup(a,y,J)∈L2H(Ω)×L2H(ΩT )×L2H(ΩT )E{〈a,−v(0)〉+ 〈u(T ), v(T )〉 − `(a, u(T ))+∫ T0〈v(t)− q1(t),−u˜(t)〉+ 〈y(t),−v˜(t)〉 − L(y(t),−u˜(t)) dt+12∫ T0〈Gv(t)−Q(t),−Fu(t)〉+ 〈J(t),−Gv(t)〉 −M(J(t),−Fu(t)) dt}.In view of the correspondence(b, r, Z) ∈ L2(Ω;H)× L2(ΩT ;H)× L2(ΩT ;H)7→ b+∫ t0r(s)ds+∫ t0Z(s)dW (s) ∈ A2H .u ∈ A2H 7→ (u(T ),−u˜,−Fu) ∈ L2(Ω;H)× L2(ΩT ;H)× L2(ΩT ;H),it follows thatL∗(q, v) = sup(a,b)∈L2H(Ω)×L2H(Ω)E{〈a,−v(0)〉+ 〈b, v(T )〉 − `(a, b)}+ supy∈L2H(ΩT )r∈L2H(ΩT )E{ ∫ T0〈v(t)− q1(t), r(t)〉+ 〈y(t),−v˜(t)〉 − L(y(t), r(t)) dt}+12supJ∈L2H(ΩT )Z∈L2H(ΩT )E{∫ T0〈Gv(t)−Q(t), Z(t)〉+ 〈J(t),−Gv(t)〉 −M(J(t), Z(t))dt},and therefore taking into account Proposition 4.1 givesL∗(q, v) = E `∗(−v(0), v(T )) + E∫ T0L∗(−v˜(t), v(t)− q1(t)) dt+12E∫ T0M∗(−Gv(t), Gv(t)−Q(t)) dt.484.3. Variational resolution of stochastic equations driven by additive noiseNow with the self-duality assumptions on L and M , and the condition on`, we have L∗(0, v) = L(v, 0), for every v ∈ A2H , which means that L ispartially self-dual on A2H × (A2H)∗.4.3 Variational resolution of stochastic equationsdriven by additive noiseFor simplicity, we shall work in an L2-setting in w and in time.4.3.1 A variational principle on Itoˆ spaceThe following is now a direct consequence of Theorem 4.1 and Theorem 3.3.Proposition 4.3. Let (Ω,F ,Ft,P) be a complete probability space with nor-mal filtration and let H be a Hilbert space. Suppose L and M are ΩT -dependent self-dual Lagrangians on H ×H, and ` is an Ω-dependent time-boundary Lagrangian on H ×H. Assume that for some positive C1, C2 andC3, we haveE∫ T0L(t, v(t), 0) dt ≤ C1(1 + ‖v‖2L2H(ΩT )) for v ∈ L2H(ΩT ),E `(a, 0) ≤ C2(1 + ‖a‖2L2H(Ω)) for a ∈ L2H(Ω),E∫ T0M(σ(t), 0) dt ≤ C3(1 + ‖σ‖2L2H(ΩT )) for σ ∈ L2H(ΩT ).(4.13)Consider on A2H the functionalI(u) = E{∫ T0(L(u(t),−u˜(t)) + 12M(Fu(t),−Fu(t)))dt+ `(u(0), u(T ))}.Then, there exists v ∈ A2H such that I(v) = infu∈A2HI(u) = 0, and consequently,P-a.s. and for almost all t ∈ [0, T ], we have− v˜(t) ∈ ∂¯L(t, v(t)) (4.14)(−v(0), v(T )) ∈ ∂`(v(0), v(T ))−Fv(t) ∈ ∂¯M(Fv(t)).Moreover, if L is strictly convex, then v is unique.494.3. Variational resolution of stochastic equations driven by additive noiseProof. The functional I can be written as I(u) = L(u, 0), where L is thepartially self-dual Lagrangian defined by (4.12).In order to apply Theorem 3.3, we need to verify the coercivity condition.To this end, we use conditions (4.13) to show that the map p → L(0, p) isbounded on the bounded sets of (A2H)∗. Indeed,L(0, p) = E{∫ T0L(t,−p1(t), 0) dt+ `(−p0, 0) + 12∫ T0M(−P (t), 0) dt}≤ C(3 + ‖p1‖2L2H(Ω) + ‖p0‖2L2H(ΩT )+ ‖P‖2L2H(ΩT )),and by duality, lim‖u‖→∞L(u, 0)‖u‖ = +∞. By Theorem 3.3, there exists v ∈ A2Hsuch that I(v) = 0. We now rewrite I as follows:0 = I(v) = E{∫ T0L(t, v(t),−v˜(t)) + 〈v(t), v˜(t)〉 dt−∫ T0〈v(t), v˜(t)〉 dt+ `(v(0), v(T )) +12∫ T0M(Fv(t),−Fv(t)) dt}.By Itoˆ’s formulaE∫ T0〈v(t), v˜(t)〉 = 12E‖v(T )‖2H −12E‖v(0)‖2H −12E∫ T0‖Fv(t)‖2H dt,which yields0 = I(v) = E{ ∫ T0(L(t, v(t),−v˜(t)) + 〈v(t), v˜(t)〉)dt}+ E{`(v(0), v(T ))− 12‖v(T )‖2H +12‖v(0)‖2H}+12E{∫ T0(‖Fv‖2H +M(Fv(t),−Fv(t)))dt}.The self-duality of the Lagrangians L and M and the hypothesis on theboundary Lagrangian, yield that for a.e. t ∈ [0, T ] and P-a.s. each of theintegrands inside the curly-brackets are non-negative, thusL(t, v(t),−v˜(t)) + 〈v(t), v˜(t)〉 = 0,`(v(0), v(T ))− 12‖v(T )‖2H +12‖v(0)‖2H = 0,M(Fv(t),−Fv(t)) + 〈Fv, Fv〉 = 0,which translate into the three assertions in (4.14).Finally, if L is strictly convex, then the functional I is strictly convex andthe minimum is attained uniquely.504.3. Variational resolution of stochastic equations driven by additive noise4.3.2 Regularization via inf-involutionThe boundedness condition (4.13) is quite restrictive and not satisfied bymost Lagrangians of interest. One way to deal with such a difficulty is toassume similar bounds on L but in stronger Banach norms. Moreover, weneed to find more regular solutions that are valued in more suitable Banachspaces than H. To this end, we consider an evolution triple V ⊂ H ⊂ V ∗,where V is a reflexive Banach space and V ∗ is its dual. We recall thefollowing easy lemma from [29].Lemma 4.2. Let L be a self-dual Lagrangian on V × V ∗.1. If for some r > 1 and C > 0, we have L(u, 0) ≤ C(1 + ‖u‖rV ) for allu ∈ V, then there exists D > 0 such that L(u, p) ≥ D(‖p‖sV ∗ − 1) forall (u, p) ∈ V × V ∗, where 1r + 1s = 1.2. If for C1, C2 > 0 and r1 ≥ r2 > 1, we haveC2(‖u‖r2V − 1) ≤ L(u, 0) ≤ C1(1 + ‖u‖r1V ) for all u ∈ V,then, there exists D1, D2 > 0 such thatD2(‖p‖s1V ∗ + ‖u‖r2V − 1) ≤ L(u, p) ≤ D1(1 + ‖u‖r1V + ‖p‖s2V ∗). (4.15)where 1ri +1si= 1 for i = 1, 2, and therefore L is continuous in bothvariables.Proposition 4.4. Consider a Gelfand triple V ⊂ H ⊂ V ∗ and let L be anΩT -dependent self-dual Lagrangian on V × V ∗. Let M be an ΩT -dependentself-dual Lagrangian on H×H, and ` an Ω-dependent boundary Lagrangianon H × H satisfying `∗(a, b) = `(−a, b). Assume the following conditionshold:(A1) For some m,n > 1, C1, C2 > 0, and for all v ∈ L2V (ΩT )C2(‖v‖mL2V (ΩT ) − 1) ≤ E∫ T0L(t, v(t), 0) dt ≤ C1(1 + ‖v‖nL2V (ΩT )).(A2) For some C3 > 0,E `(a, b) ≤ C3(1 + ‖a‖2L2H(Ω) + ‖b‖2L2H(Ω)) for all a, b ∈ L2(Ω;H).514.3. Variational resolution of stochastic equations driven by additive noise(A3) For some C4 > 0, and for all G1, G2 ∈ L2H(ΩT )E∫ T0M(G1(t), G2(t))dt ≤ C4(1 + ‖G1‖2L2H(ΩT ) + ‖G2‖2L2H(ΩT )).Then, there exists v ∈ A2H with trajectories in L2(ΩT ;V ) such that v˜ ∈L2(ΩT ;V∗), at which the minimum of the following functional is attainedand is equal to 0.I(u) = E{∫ T0(L(u(t),−u˜(t)) + 12M(Fu(t),−Fu(t)))dt+ `(u(0), u(T ))}.Consequently, P-a.s. and for almost all t ∈ [0, T ], we have− v˜(t) ∈ ∂¯L(t, v(t)) (4.16)(−v(0), v(T )) ∈ ∂`(v(0), v(T ))−Fv(t) ∈ ∂¯M(Fv(t)).Proof. First, apply Lemma 3.4 to lift L to an ΩT -dependent self-dual La-grangian onH×H, then consider for t ∈ [0, T ] and P-a.s., the λ-regularizationof L, that isLλ(u, p) = infz∈H{L(z, p) +‖u− z‖2H2λ+λ2‖p‖2H}.By Lemma 3.1, Lλ is also an ΩT -dependent self-dual Lagrangian on H ×Hin such a way that conditions (4.13) of Proposition 4.3 hold. Hence, thereexists vλ ∈ A2H such that0 = E{∫ T0Lλ(vλ(t),−v˜λ(t)) dt+ `(vλ(0), vλ(T ))+12∫ T0M(Fvλ(t),−Fvλ(t)) dt}.Since L is convex and lower semi-continuous, then dt ⊗ P a.s, there existsJλ(vλ) ∈ H so thatLλ(vλ(t),−v˜λ(t)) = L(Jλ(vλ)(t),−v˜λ(t))+‖vλ(t)− Jλ(vλ)(t)‖2H2λ+λ2‖v˜λ(t)‖2H ,524.3. Variational resolution of stochastic equations driven by additive noiseand hence0 = E{∫ T0(L(Jλ(vλ)(t),−v˜λ(t)) + ‖vλ(t)− Jλ(vλ)(t)‖2H2λ+λ2‖v˜λ(t)‖2H)dt+ `(vλ(0), vλ(T )) +12∫ T0M(Fvλ(t),−Fvλ(t)) dt}. (4.17)From (4.17), condition (A1) and the assertion of part (2) of Lemma 4.2,we can deduce that Jλ(vλ) is bounded in L2(ΩT ;V ) and v˜λ is boundedin L2(ΩT ;V∗). Also from conditions (A2) and (A3), we can deduce thefollowing estimates:E∫ T0M(G,H) dt ≥ C(‖G‖2L2H(ΩT ) − 1) and E `(a, b) ≥ C(‖b‖2L2H(Ω)− 1).These coercivity properties, together with (4.17), imply that vλ(0) and vλ(T )are bounded in L2(Ω;H), and that Fvλ is bounded in L2(ΩT ;H). Moreover,since all other terms in (4.17) are bounded below, it follows thatE∫ T0‖vλ(t)− Jλ(vλ)(t)‖2dt ≤ 2λC for some C > 0.Hence vλ is bounded inA2H and there exists a subsequence vλj that convergesweakly to a path v ∈ L2(ΩT ;V ) such that v˜ ∈ L2(ΩT ;V ∗), andJλj (vλj ) ⇀ v in L2(ΩT ;V )v˜λj ⇀ v˜ in L2(ΩT ;V∗)vλj ⇀ v in L2(ΩT ;H)vλj (0) ⇀ v(0), vλ(T ) ⇀ v(T ) in L2(Ω;H)Fvλj ⇀ Fv in L2(ΩT ;H).Since L, ` and M are lower semi-continuous, we haveI(v) ≤ lim infjE{∫ T0(L(Jλj (vλj )(t),−v˜λj (t)) +‖vλj (t)− Jλj (vλj )(t)‖22λj+λj2‖v˜λj (t)‖2)dt+ `(vλj (0), vλj (T ))+12∫ T0M(Fvλj (t),−Fvλj (t)) dt}= 0.534.3. Variational resolution of stochastic equations driven by additive noiseFor the reverse inequality, we use the self-duality of L and M and the factthat `(−a, b) = `∗(a, b) to deduce thatI(v) = E{ ∫ T0(L(v(t),−v˜(t)) + 〈v(t), v˜(t)〉)dt}+ E{`(v(0), v(T ))− 12‖v(T )‖2H +12‖v(0)‖2H}+12E{∫ T0(‖Fv‖2H +M(Fv(t),−Fv(t)))dt}≥ 0.Therefore, I(v) = 0 and the rest of the proof is similar to the last part ofthe proof in Proposition 4.3.We now deduce the following.Theorem 4.2. Consider a Gelfand triple V ⊂ H ⊂ V ∗, and let A : D(A) ⊂V → V ∗ be an ΩT -dependent progressively measurable maximal monotoneoperator satisfying〈Aw,tu, u〉 ≥ max{c1(ω, t)‖u‖αV −m1(ω, t), c2(ω, t)‖Au‖βV ∗ −m2(ω, t)},where c1, c2 ∈ L∞(ΩT , dt ⊗ P) and m1,m2 ∈ L1(ΩT , dt ⊗ P). Let B bea given H-valued progressively measurable process in L2(ΩT ;H), and u0 agiven random variable in L2(Ω,F0,P;H). Then, the equation{du(t) = −A(t, u(t))dt+B(t)dW (t)u(0) = u0,(4.18)has a solution u ∈ A2H that is valued in V . It can be obtained by minimizingthe functionalI(u) = E∫ T0L(u(t),−u˜(t)) dt+ E(12‖u(0)‖2H +12‖u(T )‖2H − 2〈u0, u(0)〉H + ‖u0‖2H)+ E∫ T0(12‖Fu(t)− 2B(t)‖2H +12‖Fu(t)‖2H − 2〈Fu(t), B(t)〉H)dt,where L is a self-dual Lagrangian such that ∂¯L(t, ·) = A(t, ·), P-almostsurely.544.4. Applications to various SPDEs with additive noiseProof. It suffices to apply Proposition 4.4 with the self-dual Lagrangian Lassociated with A, the time boundary Ω-dependent Lagrangian `u0 on H×Hgiven by`u0(a, b) =12‖a‖2H +12‖b‖2H − 2〈u0(w), a〉H + ‖u0(w)‖2H ,and the ΩT -dependent self-dual Lagrangian M on L2H(ΩT ), given byMB(G1, G2) = ΨB(w,t)(G1) + Ψ∗B(w,t)(G2),where ΨB(w,t) : H → R∪ {+∞} is the convex function ΨB(w,t)(G) = 12‖G−2B(w, t)‖2H .4.4 Applications to various SPDEs with additivenoiseIn the following examples, we shall assume D is a smooth bounded domainin Rn, W is a real Brownian motion, and B : Ω× [0, T ]→ L2(D) is a fixedprogressively measurable stochastic process.4.4.1 Stochastic evolution driven by diffusion and transportConsider the following stochastic transport equation:{du = (∆u+ a(x) · ∇u)dt+B(t)dW on [0, T ]×Du(0) = u0 on D,(4.19)where a : D → Rn is a smooth vector field with compact support in D,such that div(a) ≥ 0. Assume u0 ∈ L2(Ω,F0,P;H10 (D)) such that P-a.s.∆u0 ∈ L2(D).Consider the operator Γu = a · ∇u+ 12(div a)u, which, by Green’s formula,is skew-adjoint on H10 (D). Also consider the convex functionϕ(u) ={12∫D |∇u|2dx+ 14∫D(div a)|u|2dx u ∈ H10 (D)+∞ otherwise,which is clearly coercive on H10 (D). Consider the Gelfand triple H10 (D) ⊂L2(D) ⊂ H−1(D), and the self-dual Lagrangian on H10 (D) × H−1(D), de-fined byL(u, p) = ϕ(u) + ϕ∗(Γu+ p).554.4. Applications to various SPDEs with additive noiseThe corresponding functional on Itoˆ space is then,I(u) = E{∫ T0(ϕ(u(t, ·)) + ϕ∗ (−u˜(t, ·) + Γ(u(t, ·))))dt}+ E{12∫ T0(∫D(|Fu(t, x)|2 + 2|B(t, x)|2 − 4Fu(t, x)B(t, x))dx)dt}+ E{∫D(12|u(0, x)|2 + 12|u(T, x)|2 − 2u0(x)u(0, x) + 12|u0(x)|2)dx}.Apply Theorem 4.2 to find a path v ∈ A2L2(D), valued in H10 (D), that mini-mizes I in such a way that I(v) = 0, to obtain−v˜ + a · ∇v + 12(div a)v ∈ ∂ϕ(v) = −∆v + 12(div a)v,v(0) = u0, Fv = B.The process v(t) = u0 +∫ t0 ∆v(s)ds +∫ t0 a · ∇v(s)ds +∫ t0 B(s)dW (s) istherefore a solution to (4.19).4.4.2 Stochastic porous mediaConsider the following SPDE,{du(t) = ∆up(t)dt+B(t)dW (t) on D × [0, T ]u(0) = u0 on D,(4.20)where p ≥ n−2n+2 , and u0 ∈ L2(Ω,F0,P;H−1(D)).Equip the Hilbert space H = H−1(D) with the inner product〈u, v〉H−1 = 〈u, (−∆)−1v〉 =∫Du(x)(−∆)−1v(x) dx.Since p ≥ n−2n+2 , Lp+1(D) ⊂ H−1(D) ⊂ Lp+1p (D) is an evolution triple.We consider the convex functionalϕ(u) ={1p+1∫D |u(x)|p+1dx on Lp+1(D)+∞ elsewhere,whose Legendre conjugate is given byϕ∗(u∗) =pp+ 1∫D|(−∆)−1u∗| p+1p dx.564.4. Applications to various SPDEs with additive noiseNow, minimize the following self-dual functional on A2H ,I(u) = E{1p+ 1∫ T0∫D(|u(x)|p+1 + p ∣∣(−∆)−1(−u˜(t))∣∣ p+1p )dx dt}+ E{12‖u(0)‖2H−1+12‖u(T )‖2H−1+ ‖u0‖2H−1− 2〈u0, u(0, ·)〉H−1}+ E{∫ T012(‖Fu(t)‖2H−1+ 2‖B(t)‖2H−1− 4〈Fu(t), B(t)〉H−1)dt}.Apply Theorem 4.2 to find a process v ∈ A2H with values in Lp+1(D) suchthat(−∆)−1(−v˜(t)) ∈ ∂ϕ(v(t)) = vp, Fv = B, and v(0) = u0.It follows that v(t) = u0 +∫ t0 ∆vp(s)ds+∫ t0 B(s)dW (s), provides a solutionfor (4.20).4.4.3 Stochastic PDE involving the p-LaplacianConsider the equation{du = (∆pu− u|u|p−2)dt+B(t)dW on D × [0, T ]u(0) = u0 on ∂D,where p ∈ [2,+∞), ∆pu = div(|∇u|p−2∇u) is the p-Laplacian operator,and u0 is given such that u0 ∈ W 1,p0 (D) ∩ {u; ∆pu ∈ Lp(D)}. It is clearthat W 1,p0 (D) ⊂ Lp(D) continuously and densely, which ensures that thefunctionalϕ(u) =1p∫D|∇u(x)|pdx+ 1p∫D|u(x)|pdx,is convex, lower semi-coninuous and coercive on W 1,p0 (D) with respect tothe evolution tripleW 1,p0 (D) ⊂ Lp(D) ⊂ L2(D) ⊂W 1,p0 (D)∗ ⊂ Lq(D),where 1p +1q = 1. Theorem 4.2 applies to the self-dual functionalI(u) = E∫ T0(ϕ(t, u) + ϕ∗(t,−u˜))dt+ E(12‖u(0)‖2L2(D)+12‖u(T )‖2L2(D)− 2〈u0, u(0)〉+ ‖u0‖2L2(D))+ E∫ T0(12‖Fu(t)‖2L2(D)+ ‖B(t)‖2L2(D)− 2〈Fu(t), B(t)〉)dt.574.4. Applications to various SPDEs with additive noiseto yield a W 1,p0 (D)-valued process v ∈ A2L2(D), where the null infimum isattained. It follows that−v˜ ∈ ∂ϕ(v) = −∆pv + v|v|p−2,v(0) = u0, Fv = B,and hencev(t)− u0 −∫ t0 B(s)dW (s) =∫ t0 v˜(s)ds =∫ t0 ∆pv(s)ds−∫ t0 v(s)|v(s)|p−2ds.Remark 4.1. Note that in the examples given in the this section, the self-dual functional I is strictly convex and hence the solution obtained via min-imizing I is unique.58Chapter 5Self-dual variational principlefor stochastic partialdifferential equations withnon-additive noise5.1 IntroductionIn this chapter, we consider SPDEs driven by monotone vector fields andinvolving a non-additive noise. These can take the form{du(t) = −A(t, u(t))dt+B(t, u(t))dW (t)u(0) = u0,(5.1)where u → B(t, u) is now a progressively measurable linear or non-linearoperator. In Chapter 4, we studied the additive case of SPDEs driven bymonotone vector field, which can be formulated as completely self-dual func-tionals and in a Hilbertian setting, a solution can be obtained via a basicself-dual variational principle. In view of the evolution triple V ⊂ H ⊂ V ∗,we then applied an inf-convolution argument to find a solution that is valuedin the Sobolev space V . This approach , however, does not work in the non-additive case, since we need to work with stronger topologies on the spaceof Itoˆ processes that will give the operator B a chance to be completelycontinuous. We shall therefore strengthen the norm on the Itoˆ space over aGelfand triple, at the cost of losing coercivity, that we shall recover throughperturbation methods.In order to variationally resolve Equation (5.1), we introduce the Itoˆspace YαV analogous to A2H , but equipped with stronger norms (see Section5.2), and we show that equation{du = −∂¯L(u)(t) dt+B(t, u(t))dWu(0) = u0,595.2. Non-additive noise driven by self-dual Lagrangianshas a solution in YαV , where L is a self-dual Lagrangian on Lα(ΩT ;V ) ×Lβ(ΩT ;V∗). Taking into account the representation of maximal monotoneoperators with self-dual vector fields (Theorem 3.2) and the argument inSection 4.2.1, one can then obtain a variational resolution for Equation (5.1).The variational principle we use in this chapter is an application of The-orem 3.5 on Lα(ΩT ;V ) × Lβ(ΩT ;V ∗). However, we require to perform astochastic elliptic regularization to perturb the corresponding self-dual La-grangian so that the coercivity condition (3.8) is satisfied. Finally, we wouldlet the perturbations go to zero to conclude the existence of a solution tothe original equation. In Section 5.3, we give some immediate applicationsof the result to an SPDE driven by the gradient of a convex function, inparticular an SPDE of the form{du(t) = ∆u dt+ |u|q−1u dWu(0) = u0.where 12 ≤ q < nn−2 , and we would consider in turn a general SPDE forwhich the monotone vector field is in divergence form, namely{du = div(β(∇u(t, x)))dt+B(u(t))dW (t) in [0, T ]×Du(0, x) = u0 on ∂D,where D is a bounded domain in Rn, and β : Rn → Rn is a progressivelymeasurable maximal monotone operator.5.2 Non-additive noise driven by self-dualLagrangiansIn this section, we give a variational resolution for stochastic equations ofthe form {du = −∂¯L(u)(t) dt+B(t, u(t))dWu(0) = u0,(5.2)where L is a self-dual Lagrangian on Lα(ΩT ;V )×Lβ(ΩT ;V ∗), 1 < α < +∞and β is its conjugate, and where V ⊂ H ⊂ V ∗ is a given Gelfand triple.We shall assume that L satisfies the following conditions:C2(‖u‖αLαV (ΩT ) + ‖p‖βLβV ∗ (ΩT )− 1) ≤L(u, p) (5.3)≤ C1(1 + ‖u‖αLαV (ΩT ) + ‖p‖βLβV ∗ (ΩT )),605.2. Non-additive noise driven by self-dual Lagrangiansand‖∂¯L(u)‖LβV ∗ (ΩT )≤ C3(1 + ‖u‖α−1LαV (ΩT )). (5.4)More precisely, we are searching for a solution u of the formu(t) = u(0) +∫ t0u˜(s)ds+∫ t0Fu(s)dW (s), (5.5)where u ∈ Lα(ΩT ;V ), u˜ ∈ Lβ(ΩT ;V ∗) and Fu ∈ L2(ΩT ;H) are progressivelymeasurable. The space of such processes will be denoted by YαV and will beequipped with the norm,‖u‖YαV = ‖u(t)‖LαV (ΩT ) + ‖u˜(t)‖LβV ∗ (ΩT ) + ‖Fu(t)‖L2H(ΩT ).As shown in [48], any such a process u ∈ YαV has a dt⊗P-equivalent versionuˆ that is a V -valued progressively measurable process that satisfies the fol-lowing Itoˆ’s formula:P-a.s. and for all t ∈ [0, T ],‖u(t)‖2H = ‖u(0)‖2H + 2∫ t0〈u˜(s), uˆ(s)〉V ∗,V ds+∫ t0‖Fu(s)‖2Hds (5.6)+ 2∫ t0〈u(s), Fu(s)〉HdW (s).In particular, we have for all t ∈ [0, T ],E(‖u(t)‖2H) = E(‖u(0)‖2H) + E∫ t0(2〈u˜(s), uˆ(s)〉V ∗,V + ‖Fu(s)‖2H)ds.Furthermore, we have u ∈ C([0, T ];H). In fact, one can deduce that for anyu ∈ YαV , u ∈ C([0, T ];V ∗) and u ∈ L∞(0, T ;H) P-a.s ([44] and [48]). Fromnow on, a process u in YαV will always be identified with its dt⊗P-equivalentV -valued version uˆ.Theorem 5.1. Consider a self-dual Lagrangian L on Lα(ΩT ;V )×Lβ(ΩT ;V ∗)satisfying (5.3) and (5.4), and let B : YαV → L2(ΩT ;H) be a –not-necessarilylinear– weak-to-norm continuous map such that for some C > 0 and 0 <δ < α+12 ,‖Bu‖L2H(ΩT ) ≤ C‖u‖δLαV (ΩT )for any u ∈ YαV . (5.7)Let u0 be a given random variable in L2(Ω,F0,P;H). Equation (5.2) hasthen a solution u in YαV , that is a stochastic process satisfyingu(t) = u0 −∫ t0∂¯L(u)(s)ds+∫ t0Bu(s)dW (s). (5.8)615.2. Non-additive noise driven by self-dual LagrangiansWe would like to apply Theorem 3.5 to L on Lα(ΩT ;V ) × Lβ(ΩT ;V ∗)and to the following operators acting on G = {u ∈ YαV ;u(0) = u0},A1 : G ⊂ YαV → Lα(ΩT ;V ), Γ1 : G ⊂ YαV → Lβ(ΩT ;V ∗)A1(u) = u, Γ1(u) = −u˜A2 : G ⊂ YαV → L2(ΩT ;H), Γ2 : G ⊂ YαV → L2(ΩT ;H)A2(u) =12Fu, Γ2(u) = −Fu + 32Bu.Unfortunately, the coercivity condition (3.8) required to conclude is notsatisfied. We have to therefore perturb the Lagrangian L (i.e., essentiallyperform a stochastic elliptic regularization) as well as the operator Γ1 inorder to ensure coercivity. We will then let the perturbations go to zero toconclude.5.2.1 Stochastic elliptic regularizationTo do that, we consider the convex lower semi-continuous function on Lα(ΩT , V )ψ(u) ={1βE∫ T0 ‖u˜(t)‖βV ∗dt if u ∈ YαV+∞ if u ∈ LαV (ΩT )\YαV ,(5.9)and for any µ > 0, its associated self-dual Lagrangian on LαV (ΩT )×LβV ∗(ΩT )given byΨµ(u, p) = µψ(u) + µψ∗(pµ). (5.10)We also consider a perturbation operatorKu := (‖u‖α−1LαV (ΩT ))Du,where D is the duality map between V and V ∗. Note that by definition, Kis a weak-to-weak continuous operator from YαV to LβV ∗(ΩT ).Lemma 5.1. Under the above hypothesis on L and B, there exists a processuµ ∈ YαV such that uµ(0) = u0, u˜µ(T ) = u˜µ(0) = 0, and satisfyingu˜µ +Kuµ + µ∂ψ(uµ) ∈ −∂¯L(uµ)Fuµ = Buµ.625.2. Non-additive noise driven by self-dual LagrangiansProof. Apply Theorem 3.5 as follow: Let Z = YαV , X1 = Lα(ΩT ;V ), X2 =L2(ΩT ;H) with G = {u ∈ YαV ;u(0) = u0} which is a closed linear subspaceof YαV , and consider the operatorsA1 : G ⊂ YαV → Lα(ΩT ;V ), Γ1 : G ⊂ YαV → Lβ(ΩT ;V ∗)A1(u) = u, Γ1(u) = −u˜−KuA2 : G ⊂ YαV → L2(ΩT ;H), Γ2 : G ⊂ YαV → L2(ΩT ;H)A2(u) =12Fu, Γ2(u) = −Fu + 32Bu (5.11)where their domain is G, A1, A2 are linear, and Γ1,Γ2 are weak-weak con-tinuous.As to the Lagrangians, we take on LαV (ΩT )× LβV ∗(ΩT ), the LagrangianL1(u, p) = L ⊕Ψµ(u, p),while on L2H(ΩT )× L2H(ΩT ), we takeL2(P,Q) = E∫ T0M(P (t, w), Q(t, w)) dt,where M(P,Q) = 12‖P‖2H + 12‖Q‖2H .In other words, we are considering the functionalIµ(u) = L ⊕Ψµ(A1u,Γ1u)− E∫ T0〈A1u,Γ1u〉dt+ E∫ T0M(Γ2u,A2u)− 〈A2u,Γ2u〉 dt= L ⊕Ψµ(u,−u˜−Ku)− E∫ T0〈u,−u˜−Ku〉 dt+ E∫ T0M(Fu/2,−Fu + 3Bu/2)− 〈Fu/2,−Fu + 3Bu/2〉 dt.We now verify the conditions of Theorem 3.5.We have that G0 = Ker(A2)∩G = {u ∈ YαV ; u(t) = u0+∫ t0 u˜(s) ds}, where u˜is some progressively measurable process in LβV ∗(ΩT ). It is clear that A1(G0)is dense in Lα(ΩT ;V ). Moreover, A2(G) is dense in L2(ΩT ;H). To checkthe upper semi-continuity ofu→ E∫ T0〈A1u,Γ1u〉+ 〈A2u,Γ2u〉 dt,635.2. Non-additive noise driven by self-dual Lagrangianson YαV equipped with the weak topology, we apply Itoˆ’s formula to obtainthatE∫ T0〈A1u,Γ1u〉+ 〈A2u,Γ2u〉dt = E∫ T0〈u,−u˜−Ku〉+ 〈Fu2,−Fu + 3Bu2〉dt=12E ‖u0‖2H −12E ‖u(T )‖2H − ‖u‖α+1LαV (ΩT )+34E∫ T0〈Fu(t), Bu(t)〉dt.Upper semi-continuity then follows from the compactness of the maps YαV →L2(Ω;H) given by u 7→ (u(0), u(T )), as well as the weak to norm continuityof B, which makes the functional u 7→ E ∫ T0 〈Fu, Bu〉dt weakly continuous.To verify the coercivity, we first note first that condition (5.3) implies thatfor some (different) C1 > 0,HL(0, u) ≥ C1(‖u‖αLαV (ΩT ) − 1).By also taking into account condition (5.7) on B, with the fact that δ < α+12 ,we get thatHL(0, u) + µψ(u) + E∫ T0〈u, u˜+Ku〉 dt+ E∫ T0HM (0,Fu2)− 〈Fu2,−Fu + 3Bu2〉 dt= HL(0, u) +µβ‖u˜‖βLβV ∗ (ΩT )− 12‖u0‖2L2(Ω;H) +12‖u(T )‖2L2(Ω;H)+ ‖u‖α+1LαV (ΩT ) +18‖Fu(t)‖2L2(ΩT ;H) −34E∫ T0〈Fu(t), Bu(t)〉 dt≥ C1(‖u‖αLαV (ΩT ) − 1)+µβ‖u˜‖βLβV ∗ (ΩT )+ ‖u‖α+1LαV (ΩT )+ C2(‖Fu(t)‖2L2H(ΩT ) − ‖Fu‖L2H(ΩT )‖Bu‖L2H(ΩT ))+ C≥ C1(‖u‖αLαV (ΩT ) − 1)+µβ‖u˜‖βLβV ∗ (ΩT )+ ‖u‖α+1LαV (ΩT )+ C2(‖Fu(t)‖2L2H(ΩT ) − ‖Fu‖L2H(ΩT )‖u‖δL2H(ΩT ))+ C≥ µβ‖u˜‖βLβV ∗ (ΩT )+ ‖u‖α+1LαV (ΩT )(1 + o(‖u‖LαV (ΩT )))+ C2‖Fu(t)‖2L2H .645.2. Non-additive noise driven by self-dual LagrangiansTherefore, by Theorem 3.5, there exists uµ ∈ G ⊂ YαV such that Iµ(uµ) = 0,i.e.0 = L ⊕Ψµ(uµ,−u˜µ −Kuµ)− E∫ T0〈uµ,−u˜µ −Kuµ〉 dt+ E∫ T0M(12Fuµ ,−Fuµ +32Buµ)− 〈12Fuµ ,−Fuµ +32Buµ〉 dt.Since L ⊕ Ψµ is convex and coercive in the second variable, there existsr¯ ∈ LβV ∗(ΩT ) such thatL ⊕Ψµ(uµ,−u˜µ −Kuµ) = L(uµ, r¯) + Ψµ(uµ,−u˜µ −Kuµ − r¯),hence0 = L(uµ, r¯)− 〈uµ, r¯〉+ Ψµ(uµ,−u˜µ −Kuµ − r¯)+ E∫ T0〈uµ, u˜µ +Kuµ + r¯〉 dt+ E∫ T0M(12Fuµ ,−Fuµ +32Buµ)− 〈12Fuµ ,−Fuµ +32Buµ〉 dt.Due to the self-duality of L, Ψµ and M , this becomes the sum of threenon-negative terms, and thereforeL(uµ, r¯)− E∫ T0〈uµ(t), r¯(t)〉dt = 0,Ψµ(uµ,−u˜µ −Kuµ − r¯) + E∫ T0〈uµ(t), u˜µ(t) +Kuµ(t) + r¯(t)〉 dt = 0,E∫ T0M(12Fuµ(t),−Fuµ(t)+32Buµ(t))− 〈12Fuµ(t),−Fuµ(t) +32Buµ(t)〉 dt = 0.By the limiting case of Legendre duality, this yieldsu˜µ +Kuµ + µ∂ψ(uµ) ∈ −∂¯L(uµ) (5.12)−Fuµ(t) +32Buµ(t) ∈ ∂¯M(t, 12Fuµ(t)) =12Fuµ(t).The second line implies that for a.e. t ∈ [0, T ] we have P-a.s. Fuµ = Buµ.Moreover, from (5.12) we have that ∂ψ(uµ) ∈ LβV ∗(ΩT ).Now for an arbitrary process v ∈ YαV of the form v(t) = v(0) +∫ t0 v˜(s)ds +∫ t0 Fv(s)dW (s), we have 〈∂ψ(uµ(t)), v〉 = 〈‖u˜µ‖β−2V ∗ D−1u˜, v˜〉. Applying Itoˆ’s655.2. Non-additive noise driven by self-dual Lagrangiansformula with the progressively measurable process X(t) := ‖u˜µ‖β−2V ∗ D−1u˜,we obtainE∫ T0〈‖u˜µ‖β−2V ∗ D−1u˜µ(t), v˜(t)〉 = −E∫ T0〈 ddt(‖u˜µ‖β−2V ∗ D−1u˜µ), v(t)〉+ E 〈‖u˜µ(T )‖β−2V ∗ D−1u˜µ(T ), v(T )〉− E 〈‖u˜µ(0)‖β−2V ∗ D−1u˜µ(0), v(0)〉, (5.13)which, in view of (5.12), implies that0 = E∫ T0[〈u˜µ(t) +Kuµ(t) + ∂¯L(uµ), v〉+ µ〈‖u˜µ‖β−2V ∗ D−1u˜µ, v˜〉]dt= E∫ T0〈u˜µ(t) +Kuµ(t) + ∂¯L(uµ)− µ ddt(‖u˜µ‖β−2V ∗ D−1u˜µ), v〉dt+ µE 〈‖u˜µ(T )‖β−2V ∗ D−1u˜µ(T ), v(T )〉 − µE 〈‖u˜µ(0)‖β−2V ∗ D−1u˜µ(0), v(0)〉,hence u˜µ(T ) = u˜µ(0) = 0 and u˜µ +Kuµ − µ ddt(‖u˜µ‖β−2V ∗ D−1u˜µ) ∈ −∂¯L(uµ).In the following lemma, we shall remove the regularizing term µ∂ψ.Lemma 5.2. Under the above assumptions on L and B, there exists u ∈ YαVwith u(0) = u0, such thatL(u,−u˜−Ku) + E∫ T0〈u(t), u˜(t) +Ku(t)〉 dt = 0,Fu = Bu.Proof. Lemma 5.1 yields that for every µ > 0 there exist uµ ∈ YαV such thatuµ(0) = u0, u˜µ(T ) = u˜µ(0) = 0, and satisfyingu˜µ +Kuµ + µ∂ψ(uµ) ∈ −∂¯L(uµ) (5.14)Fuµ(t) = Buµ(t).Now we show that uµ is bounded in YαV with bounds independent of µ.Indeed, multiplying (5.14) by uµ and integrating over Ω× [0, T ], we obtainE∫ T0〈u˜µ(t) +Kuµ(t) + µ∂ψ(uµ(t)), uµ〉= −E∫ T0〈∂¯L(uµ), uµ〉dt.665.2. Non-additive noise driven by self-dual LagrangiansApply Itoˆ’s formula and use the fact that E∫ T0 〈µ∂ψ(uµ(t)), uµ〉 dt ≥ 0 toget−12‖uµ,0‖2L2(Ω;H) +12‖uµ(T )‖2L2(Ω;H) −12‖Fuµ‖2L2H(ΩT ) + ‖uµ‖α+1LαV (ΩT )= −E∫ T0〈µ∂ψ(uµ) + ∂¯L(uµ), uµ〉 dt≤ −E∫ T0〈∂¯L(uµ), uµ〉 dt.Since for uµ ∈ YαV we have uµ ∈ L∞(0, T ;H), then in view of (5.4), we getC1 + ‖uµ‖α+1LαV (ΩT ) ≤ ‖∂¯L(uµ)‖LβV ∗ (ΩT )‖uµ‖LαV (ΩT )≤ C ‖uµ‖αLαV (ΩT ).The above inequality implies that ‖uµ‖LαV (ΩT ) is bounded.Next, we multiply (5.14) by D−1u˜µ and integrate over ΩT to get that0 = E∫ T0〈u˜µ(t) +Kuµ(t) + µ∂ψ(uµ(t)) + ∂¯L(t, uµ), D−1u˜µ〉dtFrom (5.13), and choosing v = ‖u˜µ‖β−2V ∗ D−1u˜µ with v˜ = ddt(‖u˜µ‖β−2V ∗ D−1u˜µ)and Fv = 0, we get that E∫ T0 〈∂ψ(uµ(t)), D−1u˜µ〉 dt = 0, which togetherwith condition (5.4) imply that‖u˜µ‖2LβV ∗ (ΩT )≤ ‖Kuµ‖LβV ∗ (ΩT )‖u˜µ‖LβV ∗ (ΩT )+ C ‖uµ‖α−1LαV (ΩT )‖u˜µ‖LβV ∗ (ΩT ),hence‖u˜µ‖LβV ∗ (ΩT )≤ ‖Kuµ‖LβV ∗ (ΩT )+ C ‖uµ‖α−1LαV (ΩT )which means that ‖u˜µ‖LβV ∗ (ΩT )is bounded. From (5.7) and since Fuµ = Buµwe deduce that ‖Fuµ‖L2H(ΩT ) is also bounded. Now since (uµ)µ is boundedin YαV , there exists u ∈ YαV such that uµ ⇀ u weakly in YαV , which meansthat uµ ⇀ u weakly in LαV (ΩT ), u˜µ ⇀ u˜ weakly in LβV ∗(ΩT ), and Fuµ ⇀ Fuweakly in L2H(ΩT ). From (5.14) and since B is weak-norm continuous wehave Fu = Bu. Then, by (5.12) we obtain0 = L(uµ,−u˜µ −Kuµ − µ∂ψ(uµ))+ E∫ T0〈uµ(t), u˜µ(t) +Kuµ(t) + µ∂ψ(uµ(t)〉dt≥ L(uµ,−u˜µ −Kuµ − µ∂ψ(uµ)) + E∫ T0〈uµ(t), u˜µ(t) +Kuµ(t)〉 dt.675.2. Non-additive noise driven by self-dual LagrangiansSince K is weak-to-weak continuous, 〈∂ψ(uµ)), uµ〉 = ‖u˜µ‖βLβV ∗is uniformlybounded, and L is weakly lower semi-continuous on LαV × LβV ∗ , we get0 ≥ lim infµ→0L(uµ,−u˜µ −Kuµ − µ∂ψ(uµ)) + E∫ T0〈uµ(t), u˜µ(t) +Kuµ(t)〉 dt≥ L(u,−u˜−Ku) + E∫ T0〈u(t), u˜(t) +Ku(t)〉 dt.Since L is a self-dual Lagrangian on LαV × LβV ∗ , the reverse inequality isalways true, and thereforeL(u,−u˜−Ku) + E∫ T0〈u(t), u˜(t) +Ku(t)〉 dt = A general existence resultWe shall work toward eliminating the perturbation K. By Lemma 5.2, foreach ε > 0, there exists a uε ∈ G such that Fuε = Buε andL(uε,−u˜ε − εKuε) + E∫ T0〈uε(t), u˜ε(t) + εKuε(t)〉 dt = 0, (5.15)or equivalentlyu˜ε + εKuε ∈ −∂¯L(uε). (5.16)Similar to the argument in Lemma 5.2 we show that uε is bounded in YαVwith bounds independent of ε. First, we multiply (5.16) by uε and integrateover ΩT to obtainE∫ T0〈u˜ε(t) + εKuε(t), uε(t)〉 dt = −E∫ T0〈∂¯L(uε), uε〉 dt≤ ‖∂¯L(uε)‖LβV ∗ (ΩT )‖uε‖LαV (ΩT )≤ C ‖uε‖αLαV (ΩT ),where we used (5.4). In view of (5.15) and (5.3), this implies thatC(‖uε‖αLαV (ΩT ) − 1) ≤ L(uε,−u˜ε − εKuε) ≤ C ‖uε‖αLαV (ΩT ),685.3. Non-additive noise driven by monotone vector fieldsfrom which we deduce that uε is bounded in LαV (ΩT ). Next, we multiply(5.16) by D−1u˜ε to obtainE∫ T0〈u˜ε(t) + εKuε(t), D−1u˜ε(t)〉 = −E∫ T0〈∂¯L(uε), D−1u˜ε(t)〉 dt,and therefore similar to the reasoning as in Lemma 5.2 we deduce that‖u˜ε‖LβV ∗ (ΩT )≤ ε‖Kuµ‖LβV ∗ (ΩT )+ C ‖uµ‖α−1LαV (ΩT ).Hence u˜ε is bounded in LβV ∗(ΩT ), and there exists u ∈ YαV such that uε ⇀ uweakly in LαV (ΩT ), and u˜ε ⇀ u˜ weakly in LβV ∗(ΩT ), and Fuε ⇀ Fu weaklyin L2H(ΩT ). Moreover,0 = L(uε,−u˜ε − εKuε)) + E∫ T0〈uε(t), u˜ε(t) + εKuε(t)〉 dt≥ L(uε,−u˜ε − εKuε) + E∫ T0〈uε(t), u˜ε(t)〉 dt.Again, L is weakly lower semi-continuous on LαV ×LβV ∗ , therefore by lettingε→ 0 we get0 ≥ L(u,−u˜) + E∫ T0〈u(t), u˜(t)〉 dt.Since the reverse inequality is always true we haveL(u,−u˜) + E∫ T0〈u(t), u˜(t)〉 dt = 0,and also Fu(t) = Bu(t). By the limiting case of Legendre duality, we nowhave for a.e. t ∈ [0, T ], P-a.s. u˜ ∈ −∂¯L(u), integrating over [0, t] with thefact that∫ t0 u˜(s)ds = u(t) − u0 −∫ t0 Fu(s)dW (s), and Fu(t) = Bu(t) weobtainu(t) = u0 −∫ t0∂¯L(u)(s)ds+∫ t0B(u(s))dW (s).5.3 Non-additive noise driven by monotonevector fields5.3.1 Non-additive noise driven by gradient of convexenergiesThe first immediate application is the following case when the equation isdriven by the gradient of a convex function.695.3. Non-additive noise driven by monotone vector fieldsTheorem 5.2. Let V ⊂ H ⊂ V ∗ be a Gelfand triple, and let φ : V →R ∪ {+∞} be an ΩT -dependent convex lower semi-continuous function onV such that for α > 1 and some constants C1, C2 > 0, for every t ∈ [0, T ],P-a.s. we haveC2(‖u‖αLαV (ΩT ) − 1) ≤ E∫ T0φ(t, u(t)) dt ≤ C1(1 + ‖u‖αLαV (ΩT )).Consider the equation{du(t) = −∂φ(u(t))dt+B(u(t)) dW (t)u(0) = u0,(5.17)where B : YαV → L2(ΩT ;H) is a weak-to-norm continuous map satisfyingfor some C > 0 and 0 < δ < α+12 ,‖Bu‖L2H(ΩT ) ≤ C‖u‖δLα(ΩT )for any u ∈ YαV .Let u0 be a random variable in L2(Ω,F0,P;H), then Equation (5.17) has asolution u in YαV .Proof. It suffices to apply Theorem 5.1 to the self-dual LagrangianL(u, p) = E∫ T0φ(t, u(t, w)) + φ∗(t, p(t, w)) dt.Example 3. Let D ⊂ Rn be an open bounded domain, then the SPDE{du(t) = ∆u dt+ |u|q−1u dWu(0) = u0.(5.18)has a solution provided 12 ≤ q < nn−2 .Proof. Applying Theorem 5.1 with α = 2, V = H10 (D), H = L2(D), ϕ(u) =12∫D |∇u|2 dx and Bu = |u|q−1u, we see that B is weak-to-norm continuousfrom Y2V to L2(ΩT ;L2(D)), as long as 2q < 2∗, that is q < nn−2 . As to thesecond condition on B, one notes that‖Bu‖L2H(ΩT ) =(E∫ T0‖uq‖2L2(D) dt) 12≤ C‖u‖12L2V‖u‖q−12L4q−2V,705.3. Non-additive noise driven by monotone vector fieldswhich means that if 12 ≤ q ≤ 1, then 0 ≤ 4q − 2 ≤ 2 and‖Bu‖L2H(ΩT ) ≤ C‖u‖12L2V‖u‖q−12L4q−2V≤ C‖u‖12L2V‖u‖q−12L2V≤ C‖u‖qL2V,which is the condition required by the above theorem. Note that here,δ = q < 32 =α+12 .On the other hand, if 1 < q, then we apply the theorem with α = 4q − 2,then the above computation gives that‖Bu‖L2H(ΩT ) ≤ C‖u‖qL4q−2V,since 2 < 4q−2. Note also that q < 2q− 12 = α+12 . However, the Lagrangian(here the convex function ϕ) is not coercive on the space YαV = Y4q−2V . Toremedy this, we add a perturbation that makes the Lagrangian coercive onthis space by considering the convex functionϕ(u) =12∫D|∇u|2 dx+ 4q − 2∫D|∇u|4q−2 dx.By applying Theorem 5.1 with α = 4q − 2, V = W 1,α(D), H = L2(D), andϕ, we get a solution u for the equation{du(t) = (∆u+ ∆4q−2u) dt+ |u|q−1u dWu(0) = u0.(5.19)An argument like what we have already done (twice) above, then allows usto let  go to zero and get a solution for (5.18).5.3.2 Non-additive noise driven by general monotonevector fieldsMore generally, consider the following type of equations{du(t) = −A(t, u(t))dt+B(t, u(t))dW (t)u(0) = u0,(5.20)where V ⊂ H ⊂ V ∗ is a Gelfand triple, and A : Ω × [0, T ] × V → V ∗, andB : Ω× [0, T ]× V → H, are progressively measurable.Theorem 5.3. Assume A : D(A) ⊂ V → V ∗ is a progressively measurableΩT -dependent maximal monotone operator satisfying condition (4.5) withα > 1 and its conjugate β, as well as‖Aw,tu‖V ∗ ≤ k(ω, t)(1 + ‖u‖V ) for all u ∈ V , dt⊗ P a.s. (5.21)715.3. Non-additive noise driven by monotone vector fieldsfor some k ∈ L∞(ΩT ).Let B : YαV → L2(ΩT ;H) be a weak-to-norm continuous map such that forsome C > 0 and 0 < δ < α+12 ,‖Bu‖L2H(ΩT ) ≤ C‖u‖δLα(ΩT )for any u ∈ YαV .Let u0 be a given random variable in L2H(Ω,F0,P;H), then equation (5.20)has a variational solution in YαV .Proof. Associate again to Aω,t an ΩT -dependent self-dual Lagrangian LAω,ton V × V ∗ in such a way that for almost every t ∈ [0, T ], P-a.s. we haveAω,t = ∂¯LAω,t . Then by Lemma 4.1, the LagrangianLA(u, p) = E∫ T0LAω,t(u(ω, t), p(ω, t))dt,is self-dual on Lα(ΩT ;V )× Lβ(ΩT ;V ∗), and satisfiesC1(‖u‖αLαV (ΩT ) + ‖p‖βLβV ∗ (ΩT )− 1) ≤ L(u, p) ≤ C2(1 + ‖u‖αLαV (ΩT ) + ‖p‖βLβV ∗ (ΩT )).(5.21) also implies that for some C3 > 0,‖∂¯LA(u)‖LβV ∗ (ΩT )≤ C3(1 + ‖u‖LαV (ΩT )).The rest follows from Theorem Non-additive noise driven by monotone vector fieldsin divergence formWe now show the existence of a variational solution to the following equation:{du = div(β(∇u(t, x)))dt+B(u(t))dW (t) in [0, T ]×Du(0, x) = u0 on ∂D,(5.22)where D is a bounded domain in Rn, and where the initial position u0belongs to L2(Ω,F0,P;L2(D)). We assume that1. The ΩT -dependent vector field β : Rn → Rn is progressively mea-surable and maximal monotone such that for functions c1, c2, c3 ∈L∞(ΩT ), and m1,m2 ∈ L1(ΩT ), it satisfies dt⊗ P-a.s.〈β(x), x〉 ≥ max{c1‖x‖2Rn −m1, c2‖β(x)‖2Rn −m2} for all x ∈ Rn,(5.23)and‖β(x)‖Rn ≤ c3(1 + ‖x‖Rn) for all x ∈ Rn, (5.24)725.3. Non-additive noise driven by monotone vector fields2. The operator B : Y2H10 (D)→ L2(ΩT ;L2(D)) is a weak-to-norm contin-uous map such that for some C > 0 and 0 < δ < α+12 ,‖Bu‖L2L2(ΩT )≤ C‖u‖δLαH10(ΩT )for any u ∈ Y2H10 (D).Theorem 5.4. Under the above conditions on β and B, Equation (5.22)has a variational solution.We shall need the following lemma, which associates to an ΩT -dependentself-dual Lagrangian on Rn×Rn, a self-dual Lagrangian on L2(ΩT ;H10 (D))×L2(ΩT ;H−1(D)).Lemma 5.3. Let L be an ΩT -dependent self-dual Lagrangian on Rn × Rn,then the Lagrangian defined byL (u, p) = inf{E∫ T0∫DL(∇u(t, x), f(t, x)) dx dt; f ∈ L2L2Rn (D)(ΩT ),−div(f) = p}is self-dual on L2(ΩT ;H10 (D))× L2(ΩT ;H−1(D)).We shall need the following general lemma.Lemma 5.4. Let L be a self-dual Lagrangian on a Hilbert space H×H, andlet Π : V → H be a bounded linear operator from a reflexive Banach spaceV into H such that the operator Π∗Π is an isomorphism from V into V∗.Then, the LagrangianL(u, p) = inf {L(Πu, f); f ∈ H,Π∗(f) = p} ,is self-dual on V × V∗.Proof. For a fixed (q, v) ∈ V∗ × V, writeL∗(q, v) = sup{〈q, u〉+ 〈v, p〉 − L(u, p); u ∈ V, p ∈ V∗}= sup{〈q, u〉+ 〈v, p〉 − L(Πu, f); u ∈ V, p ∈ V∗, f ∈ H,Π∗(f) = p}= sup{〈q, u〉+ 〈v,Π∗f〉 − L(Πu, f); u ∈ V, f ∈ H}= sup{〈q, u〉+ 〈Πv, f〉 − L(Πu, f); u ∈ V, f ∈ H}.Since Π∗Π is an isomorphism, for q ∈ V∗ there exists a fixed f0 ∈ H suchthat Π∗f0 = q. Moreover, the spaceE = {g ∈ H; g = Πu, for some u ∈ V},735.3. Non-additive noise driven by monotone vector fieldsis closed in H in such a way that its indicator function χE on HχE(g) ={0 g ∈ E+∞ elsewhere,is convex and lower semi-continuous. Its Legendre transform is then givenfor each f ∈ H byχ∗E(f) ={0 Π∗f = 0+∞ elsewhere.It follows thatL∗(q, v) = sup{〈f0,Πu〉+ 〈Πv, f〉 − L(Πu, f); u ∈ V, f ∈ H}= sup{〈f0, g〉+ 〈Πv, f〉 − L(g, f)− χE(g); g ∈ H, f ∈ H}= (L+ χE)∗(f0,Πv)= inf{L∗(f0 − r,Πv) + χ∗E(r); r ∈ H}where we have used that the Legendre dual of the sum is inf-convolution.Finally taking into account the expression for χ∗E we obtainL∗(q, v) = inf{L∗(f0 − r,Πv); r ∈ H,Π∗r = 0}= inf{L(Πv, f0 − r); r ∈ H,Π∗r = 0}= inf{L(Πv, f); f ∈ H,Π∗f = q}= L(v, q).Proof of Lemma 5.3: This is now a direct application of Lemma 5.4.First, lift the random Lagrangian to define a self-dual Lagrangian on L2(ΩT ;L2(D;Rn))× L2(ΩT ;L2(D;Rn)), viaL(u, p) = E∫ T0∫DL(u(t, x), p(t, x)) dx dt,then use Lemma 5.4 with this Lagrangian and the operatorsL2(ΩT ;H10 (D))Π=∇−−−→ L2(ΩT ;L2(D;Rn)) Π∗=∇∗−−−−−→ L2(ΩT ;H−1(D)),745.3. Non-additive noise driven by monotone vector fieldsto get thatL is a self-dual Lagrangian on L2(ΩT ;H10 (D))×L2(ΩT ;H−1(D)).Note that Π∗Π = ∇∗∇ = −∆ induces an isomorphism from L2(ΩT ;H10 (D))to L2(ΩT ;H−1(D)).Proof of Theorem 5.4: Again, by Theorem 3.2 and the discussion inSection 4.2.1, one can associate to the maximal monotone map βω,t, anΩT -dependent self-dual Lagrangian Lβω,t(u, p) on Rn × Rn in such a waythatβω,t = ∂¯Lβω,t .If β satisfies (5.23), then the ΩT -dependent self-dual Lagrangian Lβω,t onRn × Rn satisfy for almost every t ∈ [0, T ], P-a.s.C1(‖x‖2Rn + ‖p‖2Rn − n1) ≤ Lβw,t(x, p) ≤ C2(‖x‖2Rn + ‖p‖2Rn + n2), (5.25)where C1, C2 ∈ L∞(ΩT ) and n1, n2 ∈ L1(ΩT ).We can then lift it to the space L2(ΩT ;L2Rn(D))× L2(ΩT ;L2Rn(D)) viaLβ(u, p) = E∫ T0∫DLβω,t(u(t, w, x), p(t, w, x)) dxdt,in such a way that for positive constants C1, C2 and C3 (different from above)C2(‖u‖2L2H(ΩT ) + ‖p‖2L2H(ΩT )− 1) ≤ Lβ(u, p) ≤ C1(1 + ‖u‖2L2H(ΩT ) + ‖p‖2L2H(ΩT )),where H := L2Rn(D). In view of (5.24), we also have‖∂¯Lβ(u)‖L2H(ΩT ) ≤ C3(1 + ‖u‖L2H(ΩT )).Use now Lemma 5.3 to lift Lβ to a self-dual LagrangianLβ on L2(ΩT ;H10 (D))×L2(ΩT ;H−1(D)), via the formulaLβ(u, p) = inff∈L2(ΩT ;L2Rn (D))−div(f)=p{E∫ T0∫DLβw,t(∇u(t, x), f(t, x)) dx dt;}= inf{Lβ(∇u, f); f ∈ L2(ΩT ;L2Rn(D)),−div(f) = p} . (5.26)Apply now Theorem 5.1 to get a process v ∈ Y2H10 (D)such thatLβ(v,−v˜) + 〈v, v˜〉 = 0Fv = Bv(0) = u0.755.3. Non-additive noise driven by monotone vector fieldsNow note that0 = Lβ(v,−v˜) + 〈v, v˜〉= inff∈L2(ΩT ;L2Rn (D)){E∫ T0∫DLβ(w,t)(∇v, f) dx dt; div(f) = v˜}+ E∫ T0〈v(t), v˜(t)〉H10 ,H−1dt= inff∈L2(ΩT ;L2Rn (D)){E∫ T0∫DLβ(w,t)(∇v, f)− 〈∇v(x, t), f(x, t)〉 dx dt}= inff∈L2(ΩT ;L2Rn (D))Jv(f),whereJv(f) := E∫ T0∫D{Lβ(w,t)(∇v, f)− 〈∇v(x, t), f(x, t)〉} dx dt.Note that condition (5.25) implies that L(y, 0) ≤ C(1+‖y‖2Rn), which meansthat Jv is coercive on L2(ΩT ;L2Rn(D)), thus there exists f¯ ∈ L2(ΩT ;L2Rn(D))with div(f¯) = v˜ such thatE∫ T0∫DLβ(w,t)(∇v, f¯)− 〈∇v(x, t), f¯(x, t)〉 dx dt = 0.The self-duality of L then implies that f¯(x, t) = ∂¯L(∇v(x, t)) = β(∇v(x, t)).Taking divergence leads to v˜ ∈ div (β(∇v)). Taking integrals over [0, t] andusing the fact that v ∈ Y2H10 (D)finally gives∫ t0div (β(∇v(s))) ds =∫ t0v˜(s)ds = v(t)− v(0)−∫ t0Fv(s)dW (s)= v(t)− u0 −∫ t0B(v(s))dW,which completes the proof.76Part IIEuclidean Moser-Onofriinequality and its extensionsto higher dimension77Chapter 6A dual Moser-Onofriinequality and its extensionsto higher dimension6.1 IntroductionOne of the equivalent forms of Moser’s inequality [39] on the 2-dimensionalsphere S2 states that the functionalI(u) :=14∫S2|∇u|2 dω +∫S2u dω − log(∫S2eu dω)(6.1)is bounded below on H1(S2), where dω is the Lebesgue measure on S2, nor-malized so that∫S2 dω = 1. Later, Onofri [42] showed that the infimum of(6.1) over H1(S2) is actually zero, and that modulo conformal transforma-tions, u = 0 is the only optimal function. Note that this inequality is relatedto the “prescribed Gaussian curvature” problem on S2,∆u+K(x)e2u = 1 on S2, (6.2)where K(x) is the Gaussian curvature associated to the metric g = e2ug0on S2, and ∆ = ∆g0 is the Laplace-Beltrami operator corresponding tothe standard metric g0. Finding g for a given K leads to solving (6.2).Variationally, this reduces to finding the critical points of the functionalF(u) =∫S2|∇u|2 dV04pi+ 2∫S2udV04pi− log(∫S2K(x)e2udV04pi)on H1(S2),(6.3)where the volume form is such that∫S2 dV0 = 4pi. Onofri’s result saysthat, modulo conformal transformations, u ≡ 0 is the only solution of the“prescribed Gaussian curvature” problem (6.2) for K = 1, i.e., 12∆u+eu = 1on S2, which after rescaling, u 7→ 2u, gives∆u+ e2u = 1 on S2. (6.4)786.1. IntroductionThe proof given by Onofri in [42] makes use of a constrained Moser in-equality due to Aubin [3] combined with the invariance of the functional(6.1) under conformal transformations. Other proofs were given by Osgood-Philips-Sarnak [43] and by Hong [36]. See also Ghoussoub-Moradifam [31].In this part, we use the theory of mass transport to prove that 0 is theinfimum of the functional (6.3) at least when K = 1. While this approachhas by now become standard, there are many reasons why it has not been sofar spelled out in the case of the Moser functional. The first is due to the factthat, unlike the case of Rn, optimal mass transport on the sphere is harderto work with. To avoid this difficulty, we use an equivalent formulation ofthe Onofri inequality (6.1), which is obtained by projecting (6.1) on R2 viathe stereographic projection with respect to the North pole N = (0, 0, 1),i.e., Π : S2 → R2, Π(x) :=(x11−x3 ,x21−x3)where x = (x1, x2, x3). The Moser-Onofri inequality becomes the Euclidean Onofri inequality on R2, namely116pi∫R2|∇u|2 dx+∫R2udµ2 − log(∫R2eu dµ2)≥ 0 ∀u ∈ D(R2), (6.5)where µ2 is the probability density on R2 defined by µ2(x) = 1pi(1+|x|2)2 , anddµ2 = µ2(x) dx, and D(R2) = {u ∈ L1(R2, dµ2);∇u ∈ L2(R2, dx)} .One can then try to apply the Cordero-Nazaret-Villani [15] approachas generalized by Agueh-Ghoussoub-Kang [2] and write the Energy-Entropyproduction duality for functions that are of compact support in Ω,sup{−∫Ω(F (ρ) +12|x|2ρ) dx; ρ ∈ P(Ω)}(6.6)= inf{∫Ωα|∇u|2 −G (ψ ◦ u) dx; u ∈ H10 (Ω),∫Ωψ(u) dx = 1},where G(x) = (1−n)F (x)+nxF ′(x) and where ψ and α are also computablefrom F . Here P(Ω) denotes the set of probability densities on Ω.By choosing F (x) = −nx1−1/n and ψ(t) = |t|2∗ where 2∗ = 2nn−2 andn > 2, one obtains the following duality formula for the Sobolev inequalitysup{n∫Rnρ1−1/n dx− 12∫Rn|x|2ρdx; ρ ∈ P(Rn)}= inf{2(n− 1n− 2)2 ∫Rn|∇u|2 dx;u ∈ D1,2(Rn),∫Rn|u|2∗ dx = 1},(6.7)796.1. Introductionwhere u and ρ have compact support in Rn. The extremal u∞ and ρ∞ arethen obtained as solutions of∇(|x|22− n− 1ρ1/n∞)= 0, ρ∞ = u2∗∞ ∈ P(Rn). (6.8)The best constants are then obtained by computing ρ∞ from (6.8) andinserting it into (6.7) in such a way thatinf{2(n− 1n− 2)2 ∫Rn|∇u|2 dx; u ∈ D1,2(Rn), ‖u‖2∗ = 1}= n∫Rnρ1−1/n∞ dx−12∫Rn|x|2ρ∞ dx.Note that this duality leads to a correspondence between a solution tothe Yamabe equation−∆u = |u|2∗−2u on Rn (6.9)and stationary solution to the rescaled fast diffusion equation∂tρ = ∆ρ1− 1n + div(xρ) on Rn. (6.10)The above scheme does not however apply to inequality (6.5). For one,the functions euµ2 =eu(x)pi(1+|x|2)2 do not have compact support, and if onerestricts them to bounded domains, we then need to take into considerationvarious boundary terms. What is remarkable is that a similar program canbe carried out provided the dual formula involving the free energyJΩ(ρ) = −∫Ω(F (ρ) + |x|2ρ) dxis renormalized by substituting it with JΩ(ρ)− JΩ(µ2).Another remarkable fact is that the corresponding free energy turnedout to be F (ρ) = −2ρ 12 , which is the same as the one associated to thecritical case of the Sobolev inequality F (ρ) = −nρ1− 1n when n ≥ 3. In otherwords, the Moser-Onofri inequality and the Sobolev inequality “dualize”in the same way, and both the Yamabe problem (6.9) and the prescribedGaussian curvature problem (6.4) reduce to the study of the fast diffusionequation (6.10), with the caveat that in dimension n = 2, the above equationneeds to be considered only on bounded domains, with Neumann boundary806.1. Introductionconditions.More precisely, we shall show that, when restricted to balls BR of radiusR in R2, there is a duality between the “Onofri functional”IR(u) =116pi∫BR|∇u|2 dx+∫BRu dµ2on XR := {u ∈ D(BR);∫R2eu dµ2 = 1},and the free energyJR(ρ) =2√pi∫BR√ρdx−∫BR|x|2ρdxon YR := {ρ ∈ L1+(BR);1µ2(BR)∫BRρ dx = 1,whereµ2(BR) :=∫BRdµ2 =R21 +R2.Note that if u has its support in BR, then∫R2 eu dµ2 = 1 if and only if1µ2(BR)∫BReu dµ2 = 1.We show that once the free energy is re-normalized by subtracting thefree energy of µ2, we then havesup{JR(ρ)− JR(µ2); ρ ∈ YR} = 0 = inf{IR(u); u ∈ XR}. (6.11)Note that when R→ +∞, the right hand side yields the Onofri inequal-ityinf{116pi∫R2|∇u|2 dx+∫R2udµ2; u ∈ D1,2(R2),∫R2eu dµ2 = 1}= 0,while the left-hand side doesn’t yield a universal upper bound for JR(ρ)sinceJR(µ2) = log(1 +R2) +R21 +R2→ +∞ as R→ +∞.We actually show that our approach extends to higher dimensions. Moreprecisely, if BR is a ball of radius R in Rn where n ≥ 2, and if one considersthe probability density µn on Rn defined byµn(x) =1ωn(1 + |x|nn−1 )n816.1. Introduction(ωn is the volume of the unit sphere in Rn), and the operator Hn(u, µn) onW 1,n(Rn) byHn(u, µn) := |∇u+∇(logµn)|n−|∇(logµn))|n−n|∇(logµn))|n−2∇(logµn)·∇u,there is then a duality between the functionalR(u) =1β(n)∫BRHn(u, µn) dx+∫BRudµnon XR := {u ∈ D(BR);∫Rneu dµn = 1and the free energy – renormalized by again subtracting JR(µn) –JR(ρ) = α(n)∫BRρn−1n dx−∫BR|x| nn−1 ρ dxon YR := {ρ ∈ L1+(BR);1µn(BR)∫BRρdx = 1}whereα(n) =nn− 1ωn−1/n, β(n) = ωn (nn− 1)n−1nn+1andµn(BR) :=∫BRdµn =Rn(1 +Rnn−1 )n−1.We then deduce the following higher dimensional version of the Onofriinequality: For n ≥ 2,1β(n)∫RnHn(u, µn) dx+∫Rnu dµn − log(∫Rneu dµn)≥ 0 (6.12)for all u ∈ D1,2(Rn).We finish this introduction by mentioning that there was an attempt in [18]to use mass transport to establish the Euclidean Onofri inequality (6.5) inthe radial case. In [37], Maggi and Villani also establish Sobolev-type in-equalities involving boundary trace terms via mass transport methods. Theyactually deal with a family of Moser-Trudinger inequalities as a limiting caseof Sobolev inequality when the power p → n, in the presence of boundaryterms on a Lipschitz domain in Rn. However, to our knowledge, our dualityresult, the extensions of Onofri’s inequality to higher dimensions, as well asthe mass transport proof of the general (non-radial) Onofri inequality are826.2. Preliminariesnew.In the following, first we recall the mass transport approach to sharpSobolev inequalities and some consequences. In the next section, we estab-lish the n-dimensional mass transport duality principle, from which we coulddeduce the two dimensional Euclidean Onofri inequality (6.5).6.2 PreliminariesWe start by briefly describing the mass transport approach to sharp Sobolevinequalities as proposed by [15]. We will follow here the framework of [2] asit clearly shows the correspondence between the Yamabe equation (6.9) andthe rescaled fast diffision equation (6.10).Let ρ0, ρ1 ∈ P(Rn). If T is the optimal map pushing ρ0 forward toρ1 (i.e. T#ρ0 = ρ1) in the mass transport problem for the quadratic costc(x − y) = |x−y|22 (see [52] for details), then [0, 1] 3 t 7→ ρt = (Tt)#ρ0 isthe geodesic joining ρ0 and ρ1 in (P(Rn), d2); here Tt := (1− t)id + tT andd2 denotes the quadratic Wasserstein distance (see [52]). Moreover, given afunction F : [0,∞) → R such that F (0) = 0 and x 7→ xnF (x−n) is convexand non-increasing, the functional HF (ρ) :=∫Rn F (ρ(x)) dx is displacementconvex [38], in the sense that [0, 1] 3 t 7→ HF (ρt) ∈ R is convex (in the usualsense), for all pairs (ρ0, ρ1) in P(Rn). A direct consequence is the followingconvexity inequality, known as “energy inequality”:HF (ρ1)−HF (ρ0) ≥[ddtHF (ρt)]t=0=∫Rnρ0∇(F ′(ρ0)) · (T − id) dx,which, after integration by parts of the right hand side term, reads as−HF (ρ1) ≤ −HF+nPF (ρ0)−∫Rnρ0∇(F ′(ρ0)) · T (x) dx, (6.13)where PF (x) = xF′(x)− F (x); here id denotes the identity function on Rn.By the Young inequality−∇ (F ′(ρ0))·T (x) ≤ |∇F ′(ρ0)|pp+|T (x)|qq∀p, q > 1 such that 1p+1q= 1,(6.14)(6.13) gives−HF (ρ1) ≤ −HF+nPF (ρ0) + 1p∫Rnρ0|∇F ′(ρ0)|p dx+ 1q∫Rnρ0(x)|T (x)|q dx,836.2. Preliminariesi.e.,−HF (ρ1)− 1q∫Rn|y|qρ1(y) dy ≤ −HF+nPF (ρ0) + 1p∫Rnρ0|∇F ′(ρ0)|p dx,(6.15)where we use that T#ρ0 = ρ1. Furthermore, if ρ0 = ρ1, then T = id andequality holds in (6.13). Then equality holds in (6.15) if it holds in the Younginequality (6.14). This occurs when ρ0 = ρ1 satisfies ∇(F ′ (ρ0(x)) +|x|qq)=0. Therefore, we have established the following duality:sup{−HF (ρ1)− 1q∫Rn|y|qρ1(y) dy; ρ1 ∈ P(Rn)}= inf{−HF+nPF (ρ0) + 1p∫Rnρ0|∇F ′(ρ0)|p dx; ρ0 ∈ P(Rn)},(6.16)and an optimal function in both problems is ρ0 = ρ1 := ρ∞ solution of∇(F ′ (ρ∞(x)) +|x|qq)= 0. (6.17)In particular, choosing F (x) = −nx1−1/n and ρ0 = u2∗ where 2∗ = 2nn−2and n > 2, then HF+nPF = 0, and (6.16)-(6.17) gives the duality formulafor the Sobolev inequality (6.7).Our goal now is to extend this mass transport proof of the Sobolevinequality to the Euclidean Onofri inequality (6.5). As already mentioned inthe introduction, a first attempt on this issue was recently made by [18], butthe result produced was only restricted to the radial case. Here we show infull generality (without restricting to radial functions u) that the EuclideanOnofri inequality (6.5) can be proved by mass transport techniques. Moreprecisely, we establish an analogue of the duality (6.7) for Euclidean Onofriinequality in dimension n (see Theorem (6.1)), from which we deduce then-dimensional Onofri inequality (6.35) (see Theorem 6.2). Furthermore, weobtain –as for the critical Sobolev inequality– a correspondence between theprescribed Gaussian curvature problem (6.4) and the rescaled fast diffusionequation (6.10).We shall need the following general lemma from the theory of masstransport.Lemma 6.1. Let ρ0, ρ1 ∈ P(BR), where P(BR) denotes the set of probabilitydensities on the ball BR ⊂ Rn. Let T be the optimal map pushing ρ0 forward846.3. Euclidean n-dimensional Onofri inequality: A duality formulato ρ1 (i.e. T#ρ0 = ρ1) in the mass transport problem corresponding to thequadratic cost. Then∫BRρ1(y)1− 1n dy ≤ 1n∫BRρ0(x)1− 1ndiv (T (x)) dx. (6.18)Proof. By Brenier’s theorem [10], there is a map T : BR → BR such thatT = ∇ϕ where ϕ : BR → R is convex, and T#ρ0 = ρ1. We therefore havethe following Monge-Ampe`re equation,ρ0(x) = ρ1(T (x)) det∇T (x) (6.19)or equivalentlyρ1(T (x)) = ρ0(x)[det∇T (x)]−1. (6.20)By the arithmetic-geometric-mean inequality[det∇T (x)] 1n ≤ 1ndiv(T (x)),(6.20) givesρ1(T (x))− 1n ≤ 1nρ0(x)− 1ndiv (T (x)) . (6.21)Now using the change of variable y = T (x), we have∫BRρ1(y)1− 1n dy =∫BRρ1 (T (x))1− 1n det(∇T (x)) dx,which implies by (6.19) and (6.21), that∫BRρ1(y)1− 1n dy ≤ 1n∫BRρ0(x)− 1ndiv(T (x))ρ0(x) dx=1n∫BRρ0(x)1− 1ndiv(T (x)) dx,and we are done.6.3 Euclidean n-dimensional Onofri inequality: Aduality formulaConsider the probability density on Rn, µn(y) = 1ωn(1+|y|nn−1 )n, where ωn isthe volume of the unit sphere in Rn, and setθR :=∫BRµn(y) dy =Rn(1 +Rnn−1 )n−1.We shall establish the following duality formula.856.3. Euclidean n-dimensional Onofri inequality: A duality formulaTheorem 6.1. (Duality for n-dimensional Euclidean Onofri inequal-ity) For each ball BR in Rn with radius R > 0, we consider the followingfree functionalJR(ρ) = α(n)∫BRρ(y)n−1n dy −∫BR|y| nn−1 ρ(y) dy for ρ ∈ L1+(BR),as well as the “entropy” functionalIR(u) =1β(n)∫BRHn(u, µn) dx+∫BRu(x) dµn for u ∈ D1,2(BR),whereα(n) =nn− 1(1ωn)1/n, β(n) =(nn− 1)n−1n(n+1) ωnandHn(u, µn) := |∇u+∇(logµn)|n−|∇(logµn)|n−n|∇(logµn)|n−2∇(logµn)·∇u.The following duality formula then holds:sup{JR(ρ)− JR(µn); ρ ∈ L1+(Rn),∫BRρ dy = θR}(6.22)= inf{IR(u); u ∈ D(BR),∫BReu dµn = θR}= 0.Moreover, the maximum on the l.h.s. is attained only at ρmax = µn, andthe minimum on the r.h.s. is attained only at umin = 0.Remark 6.1. Before proving the theorem, we make a few remarks on theoperator Hn(u, µn).1. Consider the function c : Rn → R, c(z) = |z|n, n ≥ 2. Clearly cis strictly convex, and ∇c(z) = n|z|n−2z. So we have the convexityinequalityc(z1)− c(z0)−∇c(z0) · (z1 − z0) ≥ 0 ∀z0, z1 ∈ Rn. (6.23)Setting z0 = ∇(logµn) and z1 = ∇u+∇(logµn), we see that Hn(u, µn)is nothing but the l.h.s of (6.23); we then deduce thatHn(u, µn) ≥ 0 ∀u, µn.866.3. Euclidean n-dimensional Onofri inequality: A duality formula2. For all u ∈ W 1,n0 (BR), the integral of Hn(u, µn) over BR involves awell-known operator, the n-Laplacian ∆n, defined by∆nv := div(|∇v|n−2∇v). (6.24)Indeed, this can be seen after performing an integration by parts in thelast term of Hn(u, µn),∫BRHn(u, µn) dy =∫BR|∇u+∇ logµn|n dy −∫BR|∇(logµn))|n dy+ n∫BRu∆n(logµn) dy.Proof. By applying Lemma 6.1, and using an integration by parts and taking1m := 1− 1n , we haven∫BRρ1/m1 dy ≤ −∫BR∇(ρ1/m0 ) · T (x) dx+∫∂BRρ1/m0 T (x) · ν dS.Use the elementary identity ∇(ρ1/m0 ) = 1mρ1/m0 ∇(log ρ0) to obtainmn∫BRρ1/m1 dy ≤ −∫BRρ1/m0 ∇(log ρ0) · T (x) dx+m∫∂Bρ1/m0 T (x) · ν dS.(6.25)Set ρ0 =euµnθR, where u ∈ D(BR) satisfies∫BReudµn = θR, and let ρ1 be anyprobability density supported on BR. By applying (6.25) to ρ0 and ρ1, wegetmn∫BR(θRρ1)1/m dy ≤−∫BR(euµn)1/m∇(log(euµn)) · T (x) dx+m∫∂BR(euµn)1/mT (x) · ν dS.Using Young’s inequality, for any ε > 0,− (euµn)1/m∇ (log(euµn)) ·T (x) ≤ 1nε|∇ (log(euµn)) |n+ εm/nmeuµn|T (x)|mand the fact that T#ρ0 = ρ1, we getmn2ε∫BR(θRρ1)1/m dy −mnε∫∂BRµ1/mn T (x) · ν dS −nmεm∫BR|y|mθRρ1 dy≤∫BR|∇u+∇ logµn|n dx.(6.26)876.3. Euclidean n-dimensional Onofri inequality: A duality formulaWe now estimate the boundary term. Since T : BR → BR, then |T (x)| ≤ Rfor all x ∈ BR∫∂BRµ1/mn T (x) · ν dS =(1ωn)1/m 1(1 +Rm)n/m∫∂BRT (x) · x|x| dS≤ nω1/nnRn(1 +Rm)n/m= nω1/nn θR.(6.27)where we used the fact that ν = x|x| .Inserting (6.27) into (6.26), and setting ρ := θRρ1, we get for all ε > 0,ε[n2m∫BRρ1/m dy − n2mω1/nn θR]− εm nm∫BR|y|mρdy (6.28)≤∫BR|∇u+∇ logµn|n dx.Now, we introduce the operator Hn(u, µn) in the r.h.s of (6.28). We have|∇u+∇ logµn|n = Hn(u, µn)+|∇(logµn))|n+n|∇(logµn))|n−2∇(logµn)·∇u,which, after an integration by parts, yields∫BR|∇u+∇ logµn|n dx =∫BRHn(u, µn) dx+∫BR|∇(logµn))|n dx− n∫BRu∆n(logµn) dx,where ∆n is the n-Laplacian operator defined by (6.24). By a direct com-putation, we note that∆n(logµn) = −nnmn−1ωnµn.It follows that∫BR|∇u+∇ logµn|n dx =∫BRHn(u, µn) dx+ nn+1mn−1ωn∫BRudµn+∫BR|∇(logµn))|n dx,and so (6.28) becomes for all ε > 0,ε[n2m∫BRρ1/m dy − n2mω1/nn θR]− εm nm∫BR|y|mρdy ≤(6.29)∫BRHn(u, µn) dx+ nn+1mn−1ωn∫BRudµn +∫BR|∇(logµn)|n dx.886.3. Euclidean n-dimensional Onofri inequality: A duality formulaNext, we focus on the l.h.s of (6.29). For convenience, we denoteAρ := n2m∫BRρ1/m dy − n2mω1/nn θR, Bρ :=nm∫BR|y|mρdy,andGρ(ε) := εAρ − εmBρ.Then for all ε > 0, (6.29) reads asGρ(ε) ≤∫BRHn(u, µn) dy + nn+1mn−1ωn∫BRudµn +∫BR|∇(logµn))|n dy.(6.30)Clearly, G′ρ(ε) = Aρ −mεm−1Bρ, so maxε>0 [Gε(ρ)] is attained atεmax(ρ) :=(AρmBρ)1/(m−1). (6.31)In particular, if ρ = µn, we haveεmax(µn) :=(AµnmBµn)1/(m−1),whereAµn = n2m(∫BRµ1/mn dy − ω1/nn θR),andBµn =nmω1/nn(∫BRµ1/mn dy − ω1/nn θR).Note that we have used above the relation∫BR|y|mµn dx =(1ωn)1/n ∫BRµ1/mn dx− θR. (6.32)Thenεmax(µn) = (nmω1/nn )1/(m−1). (6.33)Choosing ε = εmax(µn) in (6.30), we haveGρ (εmax(µn))−∫BR|∇(logµn))|n dx≤∫BRHn(u, µn) dx+ nn+1mn−1ωn∫BRudµn,896.3. Euclidean n-dimensional Onofri inequality: A duality formulathat is, after dividing by β(n) = nn+1mn−1ωn,n2mεmax(µn)β(n)∫BRρ1/m dy − n(εmax(µn))mmβ(n)∫BR|y|mρdy− 1β(n)[∫BR|∇(logµn))|n dx+ n2mω1/nn εmax(µn) θR]≤ 1β(n)∫BRHn(u, µn) dx+∫BRudµn = IR(u). (6.34)We now simplify the l.h.s of (6.34) by using the following basic identitieswhich can be checked by direct computations:n2mεmax(µn)β(n)= m(1/ωn)1/n = α(n),n(εmax(µn))mmβ(n)= 1,n2mω1/nn εmax(µn) = mnnn+1ωn,∫BR|∇(logµn)|n dx = nnmn ωn∫BR|y|mµn dy,θR =(1ωn)1/n ∫BRµ1/mn dy −∫BR|y|mµn dy.Then (6.34) yieldsJR(ρ)− JR(µn) ≤ IR(u)for all functions u and ρ such that u ∈ D(BR),∫BReu dµn = θR and∫BRρ(y) dy = θR.We conclude the proof by noting that the left-hand side is equal to 0 forρ ≡ µn, while the right-hand side is equal to 0 for u ≡ 0.From Theorem 6.1, we obtain the following n-dimensional Onofri in-equality.Theorem 6.2. (n-dimensional Euclidean Onofri inequality) For anyn ≥ 2, the following holds for any u ∈ D1,2(Rn),1β(n)∫RnHn(u, µn) dx+∫Rnu dµn − log(∫Rneu dµn)≥ 0, (6.35)hence the infimum is attained at u ≡ 0.906.3. Euclidean n-dimensional Onofri inequality: A duality formulaProof. Take u ∈ C1c (Rn) such that it has its support in a ball BR. Letv = u− C on BR and 0 elsewhere, where C is chosen so that∫BRev dµn =µn(BR). It follows that∫Rn ev dµn = 1, hence applying Theorem (6.1) weget thatIR(v) =1β(n)∫BRHn(v, µn) dx+∫BRv dµn(x)− log∫Rnev dµn ≥ 0. (6.36)Since Hn(v, µn) = Hn(u, µn), then (6.36) gives1β(n)∫RnHn(u, µn) dx+∫Rnu(x) dµn(x)− log(∫Rneu dµn)≥ 0.From the proof of Theorem 6.1, we can also derive the following inequality.Corollary 6.1. Let n ≥ 2 be an integer. For v ∈ C1c (Rn) with compactsupport in BR ⊂ Rn for some R > 0, we have(1ωn)n−1n 1(1 +Rnn−1 )n∫BRev dx+n− 1n2∫BR|∇v|n dx ≥∫BRµn−1nn dy.(6.37)In particular, if n = 2, then (6.37) gives∫BRev dx+(1 +R2)2√pi4∫BR|∇v|2 dx ≥ pi(1 +R2)2 log(1 +R2). (6.38)Proof. Choosing ρ = µn and ε = εmax(µn) in (6.28), we haveGµn(εmax(µn)) ≤∫BR|∇(u+ logµn)|n dx, (6.39)for any u such that∫BReu dµn = θR and u|∂BR = 0. Using the computationsin the proof of Theorem 6.1 and setting m := nn−1 , we haveGµn(εmax(µn)) =Anµnn(mBµn)n−1 = mn(∫BRµ1/mn dy − ω1/nn θR).This givesmn(∫BRµ1/mn dy − ω1/nn θR)≤∫BR|∇(u+ logµn)|n dx.916.3. Euclidean n-dimensional Onofri inequality: A duality formulaSet v := u+ logµn − log (µn|∂BR). We have∇v = ∇(u+ logµn), v|∂BR = 0, θR =∫BReu dµn = µn|∂BR∫BRev dx,where µn|∂BR = 1ωn(1+Rm)n . Then (6.39) reads asmn(∫BRµ1/mn dy − ω1/nn1ωn(1 +Rm)n∫BRev dx)≤∫BR|∇v|n dx. (6.40)This gives (6.37) after simplification. Using∫BR√µn =√pi log(1 + R2)where BR ⊂ R2, we get (6.38).In dimension n = 2, the operator Hn becomes H2(u, µ2) = |∇u|2, andTheorem (6.1) then yields the 2-dimensional Onofri inequality.Corollary 6.2. (Duality for the 2-dimensional Euclidean Onofri in-equality) For any ball BR of radius R > 0 in R2, consider the functionalsIR(u) =116pi∫BR|∇u(x)|2 dx+∫BRu(x) dµ2(x) on C∞c (BR),andJR(ρ) =2√pi∫BR√ρ(y) dy −∫BR|y|2ρ(y) dy on L1+(R2).1. 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