Nonsmooth variational problems and their inequalities: comparison principles and applications
Siegfried Carl, Vy K. Le, Dumitru MotreanuThis monograph focuses primarily on nonsmooth variational problems that arise from boundary value problems with nonsmooth data and/or nonsmooth constraints, such as is multivalued elliptic problems, variational inequalities, hemivariational inequalities, and their corresponding evolution problems.
The main purpose of this book is to provide a systematic and unified exposition of comparison principles based on a suitably extended subsupersolution method. This method is an effective and flexible technique to obtain existence and comparison results of solutions. Also, it can be employed for the investigation of various qualitative properties, such as location, multiplicity and extremality of solutions. In the treatment of the problems under consideration a wide range of methods and techniques from nonlinear and nonsmooth analysis is applied, a brief outline of which has been provided in a preliminary chapter in order to make the book selfcontained.
This text is an invaluable reference for researchers and graduate students in mathematics (functional analysis, partial differential equations, elasticity, applications in materials science and mechanics) as well as physicists and engineers.
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Springer Monographs in Mathematics Siegfried Carl Vy Khoi Le Dumitru Motreanu Nonsmooth Variational Problems and Their Inequalities Comparison Principles and Applications Siegfried Carl Institut für Mathematik MartinLutherUniversität HalleWittenberg D06099 Halle Germany siegfried.carl@mathematik.unihalle.de Vy Khoi Le Department of Mathematics and Statistics University of MissouriRolla Rolla, MO 65409 U.S.A vy@umr.edu Dumitru Motreanu Département de Mathématiques Université de Perpignan 66860 Perpignan France motreanu@univperp.fr Mathematics Subject Classifications (2000): (Primary) 35B05, 35J20, 35J85, 35K85, 35R70, 47J20, 47J35, 49J52, 49J53; (Secondary) 35J60, 35K55, 35R05, 35R45, 49J40, 58E35 Library of Congress Control Number: 2006933727 ISBN13: 9780387306537 eISBN13: 9780387462523 Printed on acidfree paper. © 2007 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science +Business Media LLC, 233 Spring Street, New York, NY 10013, U.S.A.), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 springer.com (TXQ/SB) Preface Nonsmooth variational problems have their origin in the study of nondiﬀerentiable energy functionals, and they arise as necessary conditions of critical points of such functionals. In this way, variational inequalities are related with convex energy or potential functionals, whereas the new class of hemivariational inequalities arise in the study of nonconvex; potential functionals that are, in general, merely locally Lipschitz. The foundation of variational inequalities is from Fichera, Lions, and Stampacchia, and it dates back to the 1960s. Hemivariational inequalities were ﬁrst introduced by Panagiotopoulos about two decades ago and are closely related with the development of the new concept of Clarke’s generalized gradient. By using this new type of inequalities, Panagiotopoulos was able to solve various open questions in mechanics and engineering. This book focuses on nonsmooth variational problems not necessarily related with some potential or energy functional, which arise, e.g., in the study of boundary value problems with nonsmooth data and/or nonsmooth constraints such as multivalued elliptic problems with multifunctions of Clarke’s subgradient type, variational inequalities, hemivariational inequalities, and their corresponding evolutionary counterparts. The main purpose is to provide a systematic and uniﬁed exposition of comparison principles based on a suitably extended subsupersolution method. This method manifests as an eﬀective and ﬂexible technique to obtain existence and comparison results of solutions. Moreover, it can be employed for the investigation of various qualitative properties such as location, multiplicity, and extremality of solutions. In the treatment of the problems under consideration, a wide range of methods and techniques from nonlinear and nonsmooth analysis are applied; a brief outline of which has been provided in a preliminary chapter to make the book selfcontained. The book is an outgrowth of the authors’ research on the subject during the past 10 years. A great deal of the material presented here has been obtained only in recent years and appears for the ﬁrst time in book form. vi Preface The materials presented in our book are accessible to graduate students in mathematical and physical sciences, researchers in pure and applied mathematics, physics, mechanics, and engineering. It is our pleasure to acknowledge a debt of gratitude to Dr. Viorica Motreanu for her competent and dedicated help during the preparation of this book at its various stages. Finally, the authors are grateful to the very professional editorial staﬀ of Springer, particularly to Ana Bozicevic and Vaishali Damle for their eﬀective and productive collaboration. Halle Rolla Perpignan September 2005 Siegfried Carl Vy K. Le Dumitru Motreanu Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Operators in Normed Linear Spaces . . . . . . . . . . . . . . . . . 2.1.2 Duality in Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Convex Analysis and Calculus in Banach Spaces . . . . . . 2.1.4 Partially Ordered Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Spaces of Lebesgue Integrable Functions . . . . . . . . . . . . . . 2.2.2 Deﬁnition of Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Chain Rule and Lattice Structure . . . . . . . . . . . . . . . . . . . 2.2.4 Some Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Operators of Monotone Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Main Theorem on Pseudomonotone Operators . . . . . . . . 2.3.2 Leray–Lions Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Multivalued Pseudomonotone Operators . . . . . . . . . . . . . . 2.4 FirstOrder Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 VectorValued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Evolution Triple and Generalized Derivative . . . . . . . . . . 2.4.4 Existence Results for Evolution Equations . . . . . . . . . . . . 2.4.5 Multivalued Evolution Equations . . . . . . . . . . . . . . . . . . . . 2.5 Nonsmooth Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Clarke’s Generalized Gradient . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Some Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Critical Point Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Linking Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 11 11 15 20 27 28 28 30 34 36 39 39 41 45 49 50 53 55 59 62 63 63 68 73 77 viii Contents 3 Variational Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.1 Semilinear Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.1.1 Comparison Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.1.2 Directed and Compact Solution Set . . . . . . . . . . . . . . . . . . 84 3.1.3 Extremal Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.2 Quasilinear Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.2.1 Comparison Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.2.2 Directed and Compact Solution Set . . . . . . . . . . . . . . . . . . 97 3.2.3 Extremal Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.3 Quasilinear Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.3.1 Parabolic Equation with pLaplacian . . . . . . . . . . . . . . . . . 110 3.3.2 Comparison Principle for Quasilinear Equations . . . . . . . 112 3.3.3 Directed and Compact Solution Set . . . . . . . . . . . . . . . . . . 116 3.3.4 Extremal Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.4 SignChanging Solutions via Fučik Spectrum . . . . . . . . . . . . . . . 123 3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 3.4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.4.3 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 3.5 Quasilinear Elliptic Problems of Periodic Type . . . . . . . . . . . . . . 134 3.5.1 Problem Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 3.5.2 SubSupersolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.5.3 Existence Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 3.6 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4 Multivalued Variational Equations . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.1 Motivation and Introductory Examples . . . . . . . . . . . . . . . . . . . . . 143 4.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4.1.2 Comparison Principle: Subdiﬀerential Case . . . . . . . . . . . 146 4.1.3 Comparison Principle: Clarke’s Gradient Case . . . . . . . . . 149 4.2 Inclusions with Global Growth on Clarke’s Gradient . . . . . . . . . 155 4.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.2.2 Comparison and Compactness Results . . . . . . . . . . . . . . . 160 4.3 Inclusions with Local Growth on Clarke’s Gradient . . . . . . . . . . 167 4.3.1 Comparison Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 4.3.2 Compactness and Extremality Results . . . . . . . . . . . . . . . 176 4.4 Application: Diﬀerence of Multifunctions . . . . . . . . . . . . . . . . . . . 180 4.4.1 Hypotheses and Main Result . . . . . . . . . . . . . . . . . . . . . . . . 181 4.4.2 A Priori Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 4.4.3 Proof of Theorem 4.36 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 4.5 Parabolic Inclusions with Local Growth . . . . . . . . . . . . . . . . . . . . 190 4.5.1 Comparison Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 4.5.2 Extremality and Compactness Results . . . . . . . . . . . . . . . 201 4.6 An Alternative Concept of SubSupersolutions . . . . . . . . . . . . . . 208 4.7 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Contents ix 5 Variational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 5.1 Variational Inequalities on Closed Convex Sets . . . . . . . . . . . . . . 213 5.1.1 Solutions and Extremal Solutions above Subsolutions . . 214 5.1.2 Comparison Principle and Extremal Solutions . . . . . . . . . 226 5.2 Variational Inequalities with Convex Functionals . . . . . . . . . . . . 234 5.2.1 General Settings—Sub and Supersolutions . . . . . . . . . . . 235 5.2.2 Existence and Comparison Results . . . . . . . . . . . . . . . . . . . 238 5.2.3 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 5.3 Evolutionary Variational Inequalities . . . . . . . . . . . . . . . . . . . . . . . 246 5.3.1 General Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 5.3.2 Comparison Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 5.3.3 Obstacle Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 5.4 SubSupersolutions and Monotone Penalty Approximations . . . 257 5.4.1 Hypotheses and Preliminary Results . . . . . . . . . . . . . . . . . 258 5.4.2 Obstacle Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 5.4.3 Generalized Obstacle Problem . . . . . . . . . . . . . . . . . . . . . . 262 5.5 Systems of Variational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 267 5.5.1 Notations and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 268 5.5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 5.5.3 Comparison Principle for Systems . . . . . . . . . . . . . . . . . . . 272 5.5.4 Generalization, Minimal and Maximal Solutions . . . . . . . 274 5.5.5 Weakly Coupled Systems and Extremal Solutions . . . . . 275 5.6 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 6 Hemivariational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 6.1 Notion of SubSupersolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 6.2 Quasilinear Elliptic Hemivariational Inequalities . . . . . . . . . . . . . 285 6.2.1 Comparison Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 6.2.2 Extremal Solutions and Compactness Results . . . . . . . . . 290 6.2.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 6.3 Evolutionary Hemivariational Inequalities . . . . . . . . . . . . . . . . . . 299 6.3.1 SubSupersolutions and Equivalence of Problems . . . . . . 301 6.3.2 Existence and Comparison Results . . . . . . . . . . . . . . . . . . . 303 6.3.3 Compactness and Extremality Results . . . . . . . . . . . . . . . 310 6.4 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 7 Variational–Hemivariational Inequalities . . . . . . . . . . . . . . . . . . . 319 7.1 Elliptic Variational–Hemivariational Inequalities . . . . . . . . . . . . . 319 7.1.1 Comparison Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 7.1.2 Compactness and Extremality . . . . . . . . . . . . . . . . . . . . . . 328 7.2 Evolution Variational–Hemivariational Inequalities . . . . . . . . . . . 336 7.2.1 Deﬁnitions and Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . 338 7.2.2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 7.2.3 Existence and Comparison Result . . . . . . . . . . . . . . . . . . . 343 7.2.4 Compactness and Extremality . . . . . . . . . . . . . . . . . . . . . . 351 x Contents 7.3 Nonsmooth Critical Point Theory . . . . . . . . . . . . . . . . . . . . . . . . . 355 7.4 A Constraint Hemivariational Inequality . . . . . . . . . . . . . . . . . . . . 362 7.5 Eigenvalue Problem for a Variational–Hemivariational Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 7.6 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 1 Introduction A powerful and fruitful tool for proving existence and comparison results for a wide range of nonlinear elliptic and parabolic boundary value problems is the method of sub and supersolutions. In one of its simplest forms, this method is a consequence of the classic maximum principle for sub and superharmonic functions that can be seen in the following classic example. Consider the homogeneous Dirichlet boundary value problem −Δu = f in Ω, u=0 on ∂Ω, (1.1) where Ω ⊂ RN is a bounded domain with smooth boundary ∂Ω, f : Ω → R is some given smooth function, and assume the existence of a classic subsolution u and supersolution ū of (1.1), i.e., u, ū ∈ C 2 (Ω) ∩ C(Ω) satisfying −Δu ≤ f in Ω, u≤0 on ∂Ω, (1.2) −Δū ≥ f in Ω, ū ≥ 0 on ∂Ω. (1.3) Then w = u−ū is readily seen as a subharmonic function in Ω with nonpositive boundary values, i.e., −Δw ≤ 0 in Ω, w≤0 on ∂Ω, (1.4) and thus, by the classic maximum principle (see [187]), it follows that w ≤ 0 in Ω, i.e., u ≤ ū in Ω. Moreover, because any solution u of (1.1) satisﬁes both (1.2) and (1.3), it must be at the same time a subsolution and a supersolution of (1.1), which implies the unique solvability of the Dirichlet problem (1.1). Thus, in view of the maximum principle, any pair of subsupersolutions of (1.1) must be ordered, and the solution u of (1.1) must be unique and must be contained in the ordered interval [u, ū]. In this way, the maximum principle enables us to obtain a priori bounds for the solution of problem (1.1). Also, an immediate consequence of the maximum principle is the orderpreserving property of solutions of (1.1), which means that if u1 and u2 are the solutions 2 1 Introduction of (1.1) corresponding to righthand sides f1 and f2 , respectively, satisfying f1 ≤ f2 , then u1 ≤ u2 in Ω. Unfortunately, maximum principles do not hold in many nonlinear elliptic problems written in the abstract form Au = f in Ω, Bu = 0 on ∂Ω. (1.5) However, if u and ū are appropriate (weak) sub and supersolutions of (1.5) satisfying, in addition, u ≤ ū, then (weak) solutions of (1.5) (not necessarily unique) exist within the interval [u, ū] formed by the ordered pair of sub and supersolutions. It is basically this property that we will refer to as a comparison principle for the problems under consideration. For example, consider the following prototype of (1.5): −Δp u + g(u) = f in Ω, u=0 on ∂Ω, (1.6) where Δp u = div (∇up−2 ∇u) is the pLaplacian, 1 < p < ∞, f ∈ Lq (Ω) with q being the Hölder conjugate to p satisfying 1/p + 1/q = 1, and g : R → R is a continuous function with some growth condition. As is well known, in general, problem (1.6) does not admit classic solutions, and therefore, it has to be treated within the framework of weak solutions. Let V = W 1,p (Ω) and V0 = W01,p (Ω) denote the usual Sobolev spaces with their dual spaces V ∗ and V0∗ , respectively, then a weak solution of the Dirichlet problem (1.6) is deﬁned as follows: u ∈ V0 : −Δp u + g(u) = f in V0∗ , (1.7) where due to the continuous embedding Lq (Ω) ⊂ V0∗ , f has to be interpreted as a dual element of V0∗ . As Au = −Δp u + g(u) deﬁnes a bounded and continuous mapping from V0 into V0∗ , (1.7) provides an appropriate functional analytic framework for the boundary value problem (1.6), which is equivalent with the following variational equation: u ∈ V0 : −Δp u + g(u), ϕ = f, ϕ for all ϕ ∈ V0 , (1.8) where ·, · denotes the duality pairing. It follows from standard integration by parts that the variational equation (1.8) is equivalent to u ∈ V0 : ∇up−2 ∇u∇ϕ dx + g(u) ϕ dx = f, ϕ for all ϕ ∈ V0 . Ω Ω (1.9) A natural extension of the classic notion of sub and supersolution to the weak formulation (1.7) of the boundary value problem (1.6) is deﬁned as follows. The function ū ∈ V is a weak supersolution of (1.7) if ū ≥ 0 on ∂Ω and − Δp ū + g(ū) ≥ f in V0∗ , (1.10) 1 Introduction 3 where the inequality in V0∗ has to be taken with respect to the dualorder cone ∗ of V0∗ , deﬁned by V0,+ ∗ V0,+ = {u∗ ∈ V0∗ : u∗ , ϕ ≥ 0 for all ϕ ∈ V0 ∩ Lp+ (Ω)}, where Lp+ (Ω) is the positive cone of all nonnegative elements of Lp (Ω) by which the natural partial ordering of functions in Lp (Ω) is deﬁned. Due to (1.10), we obtain the following wellknown equivalent deﬁnition of a weak supersolution of (1.7). The function ū ∈ V is a weak supersolution if ū ≥ 0 on ∂Ω and ∇ūp−2 ∇ū∇ϕ dx + g(ū) ϕ dx ≥ f, ϕ for all ϕ ∈ V0 ∩ Lp+ (Ω). Ω Ω (1.11) Similarly, u ∈ V is a weak subsolution of (1.7) if u ≤ 0 on ∂Ω and p−2 ∇u ∇u∇ϕ dx + g(u) ϕ dx ≤ f, ϕ for all ϕ ∈ V0 ∩ Lp+ (Ω). Ω Ω (1.12) Comparison principles for solutions of nonlinear elliptic and parabolic variational equations including the special case (1.7) are well known and can be found, e.g., in the monographs [43, 66, 83]. Thus, we have, e.g., if u and ū are sub and supersolutions of (1.7), respectively, and if u ≤ ū, then solutions exist within the ordered interval [u, ū]. Moreover, the solution set S enclosed by an ordered pair of sub and supersolutions can be shown to be compact and to possess greatest and smallest elements with respect to the natural partial ordering of functions induced by the order cone Lp+ (Ω). A review and detailed proofs of these results will be given in Chap. 3. The existence and comparison results along with the topological and order related characterization of the solution set S obtained for nonlinear elliptic and parabolic variational equations generalize the following elementary result on the real line R. Consider the real equation F (u) = 0, u ∈ R, (1.13) and assume that: (i) The function F : R → R is continuous. (ii) s, s̄ ∈ R satisfying s ≤ s̄ exist such that F (s) ≤ 0 and F (s̄) ≥ 0. Then solutions of (1.13) exist within the real interval [s, s̄], and the set of all solutions of (1.13) is closed and bounded and, thus, compact. Moreover, the solution set has a greatest and smallest element s∗ and s∗ , respectively (see Fig. 1.1). This classic existence and enclosure result follows from the intermediate value theorem for continuous functions, whereas the existence of greatest and 4 1 Introduction Fig. 1.1. Subsupersolution smallest solutions is an immediate consequence of the order property of the real line R, which, speaking in abstract terms, is a completely ordered Banach space. Now the results above concerning (weak) solutions of the nonlinear problem (1.6) nicely ﬁt into this elementary picture. Let F : V0 → V0∗ be deﬁned by F (u) = −Δp u + g(u) − f. Then the equivalent elliptic variational equation (1.7) can be rewritten as u ∈ V0 : F (u) = 0 in V0∗ . Assume that: (i∗ ) The function g : R → R is continuous and satisﬁes a certain growth condition. (ii∗ ) u, ū ∈ V satisfying u ≤ ū exist with u ≤ 0 on ∂Ω, ū ≥ 0 on ∂Ω such that F (u) ≤ 0 and F (ū) ≥ 0. Then the existence and comparison result as well as the characterization of the solution set for (1.6) given above hold. Note that in view of (i∗ ), the operator F : V0 → V0∗ is continuous, bounded, and pseudomonotone, but not necessarily coercive. As will be seen in Chap. 3, the existence of sub and supersolutions supposed in (ii∗ ) will be used to compensate this drawback. In this monograph, we focus primarily on nonsmooth variational problems. Just as “nonlinear” in mathematics stands for “not necessarily linear,” we use 1 Introduction 5 “nonsmooth” to refer to certain situations in which smoothness is not necessarily assumed. The relaxed smoothness requirements have often been motivated by the needs of disciplines other than mathematics, such as mechanics and engineering. Our main goal is to extend the idea of subsupersolutions and to provide a systematic and uniﬁed approach for obtaining comparison principles for both nonsmooth stationary and evolutionary variational problems. We shall demonstrate that much of the idea of the method of subsupersolutions that has been known for elliptic and parabolic variational equations can be developed in a general nonsmooth setting. To give an idea of what we mean by nonsmooth variational problems, let us consider a few examples. A nonsmooth variational problem arises, e.g., when the nonlinearity g in (1.9) is no longer continuous. If g : R → R satisﬁes some growth condition but is only supposed to be Borelmeasurable, then problem (1.9) becomes a discontinuous variational equation. Even though the operator A of the equivalent operator equation (1.7) given by Au = −Δp u + g(u) is still well deﬁned and bounded from V0 into its dual space V0∗ ; it is, however, no longer continuous. In this case, the subsupersolution method, in general, fails as shown by the following simple example. Let us consider (1.7) with p = 2, f (x) ≡ 1, and g the Heaviside step function given by g(s) = 0 for s ≤ 0, and g(s) = 1 for s > 0; i.e., we consider u ∈ V0 = W01,2 (Ω) : −Δu + g(u) = 1 in V0∗ . (1.14) One readily veriﬁes that the constant functions u = −c and ū = c with c any positive constant provide an ordered pair of subsupersolutions of (1.14). However, problem (1.14) has no solutions within the order interval [−c, c]. Furthermore, (1.14) does not possess solutions at all. In fact, if u was a solution, then it satisﬁes the variational equation ∇u∇ϕ dx = (1 − g(u)) ϕ dx for all ϕ ∈ V0 . Ω Ω Taking as a special test function the solution u, we obtain in view of the deﬁnition of g the following inequality: 2 ∇u dx = (1 − g(u)) u dx ≤ 0, Ω Ω and hence it follows that u = 0. This result is a contradiction, because u = 0 is apparently not a solution of (1.14). Problem (1.14) with g being the Heaviside function is embedded into a relaxed multivalued setting replacing the discontinuous function g by an associated multivalued function s → [g(s), ḡ(s)], where g(s) and ḡ(s) denote the leftsided and rightsided limits of g at s ∈ R. It turns out that this multifunction that, roughly speaking, arises from g by ﬁlling in the gap at the point of discontinuity, coincides with the multifunction s → ∂j(s), where ∂j(s) denotes 6 1 Introduction Fig. 1.2. Subdiﬀerential of j Fig. 1.3. Primitive of Heaviside function u the subdiﬀerential of the primitive j : R → R of g given by j(u) = 0 g(s) ds, which is a convex and Lipschitz continuous function, (see Fig. 1.2 and Fig. 1.3). Thus, the relaxed multivalued problem (1.14) reads as follows: u ∈ V0 : −Δp u + ∂j(u) 1 in V0∗ , (1.15) where j : R → R is the above primitive of the Heaviside function. As j is convex and even Lipschitz continuous, one can easily show that (1.15) is equivalent to u ∈ V0 : ∂ Ê(u) 0, where ∂ Ê(u) is the subdiﬀerential at u of the nonsmooth, convex, continuous, and coercive functional Ê : V0 → R deﬁned by 1 p Ê(u) = ∇u dx + j(u) dx − 1, u . p Ω Ω As Ê in our example is even strictly convex, a unique solution of the optimization problem exists u ∈ V0 : Ê(u) = inf Ê(v), v∈V0 which in turn is equivalent to ∂ Ê(u) 0. Thus, problem (1.15) has only one solution, which is the minimum point of the nonsmooth functional Ê. To motivate other types of nonsmooth variational problems, consider the functional E: 1 E(u) = ∇up dx + j(u) dx − f, u , u ∈ V0 , (1.16) p Ω Ω 1 Introduction 7 where f ∈ V0∗ and j : R → R is the primitive of a continuous function g that satisﬁes some growth condition. Then E : V0 → R is a C 1 functional whose critical points are the solutions of the variational problem (1.9). In this sense, (1.9) may be considered as a smooth variational problem in case g is continuous. A nonsmooth variational problem already occurs if we are looking for critical points of the C 1 functional E of (1.16) under some constraint, which is represented, for example, by a closed convex subset K ⊂ V0 . This leads to the following wellknown variational inequality for the operator Au = −Δp u+ g(u): u∈K: Au − f, ϕ − u ≥ 0, for all ϕ ∈ K. (1.17) Introducing the indicator function IK of the set K, we see that (1.17) is equivalent to the variational inequality u∈K: Au − f, ϕ − u + IK (ϕ) − IK (u) ≥ 0, for all ϕ ∈ V0 , (1.18) which in turn is equivalent to the diﬀerential inclusion u∈K: −Au + f ∈ ∂IK (u), where ∂IK is the subdiﬀerential of the indicator function IK : V0 → [0, +∞], which is proper if K = ∅, convex, and lower semicontinuous. Another type of nonsmooth variational problems arises if we consider critical points of the functional E above when j is the primitive of a not necessarily continuous function g satisfying only some growth and measurability conditions. Under these assumptions, E : V0 → R is, in general, no longer convex, but only locally Lipschitz, and u is called a critical point of E if 0 ∈ ∂E(u), (1.19) where ∂E(u) ⊂ V0∗ denotes Clarke’s generalized gradient. For example, if u is a minimum point of E over V0 , then u is a critical point, and it satisﬁes (1.19). Applying basic facts from nonsmooth analysis, we see that (1.19) is equivalent to u ∈ V0 : −Δp u − f, ϕ + J o (u; ϕ) ≥ 0, for all ϕ ∈ V0 , (1.20) where J o (u; v) denotes the generalized directional derivative at u in direction v of the locally Lipschitz functional J : V0 → R given by J(u) = j(u) dx. Ω Problem (1.20) is called a hemivariational inequality, which is equivalent to the inclusion Δp u + f ∈ ∂J(u), 8 1 Introduction Fig. 1.4. Zigzag friction law where ∂J(u) ⊂ V0∗ is Clarke’s generalized gradient of the integral functional J at u. Closely related but, in general, not equivalent to (1.20) is the following diﬀerential inclusion: u ∈ V0 : −Δp u + ∂j(u) f in V0∗ , (1.21) where ∂j : R → 2R \{∅} is Clarke’s generalized gradient of the locally Lipschitz integrand j : R → R of J. An example of multifunctions ∂j that appear in applications is shown in Fig. 1.4. Finally, if we try to ﬁnd solutions of the hemivariational inequality under constraints, we arrive at the nonsmooth variational problem u∈K: −Δp u − f, ϕ − u + IK (ϕ) − IK (u) + J o (u; ϕ − u) ≥ 0, for all ϕ ∈ V0 , (1.22) which is called a variationalhemivariational inequality. The ﬁeld of hemivariational inequalities, initiated with the pioneering work of Panagiotopoulos (cf. [179, 180]), has attracted increasing attention over the last decade mainly due to its many applications in mechanics and engineering. This new type of variational inequalities arises, e.g., in mechanical problems governed by nonconvex, possibly nonsmooth energy functionals (socalled superpotentials), which appear if nonmonotone, multivalued constitutive laws are taken into account. However, note that the multivalued problems (1.15) and (1.21), the variational inequality (1.17), the hemivariational inequality (1.20), and the variationalhemivariational inequality (1.22) only serve as prototypes of nonsmooth variational problems of elliptic type that will be treated in this book. Comparison principles will be obtained for more general nonsmooth variational problems that are not necessarily related to some potential functional, and for 1 Introduction 9 their evolutionary counterparts. It should be noted also that the treatment of evolutionary nonsmooth variational problems is by no means a straightforward extension of nonsmooth (stationary) elliptic variational problems, and it requires diﬀerent tools. Moreover, not only scalar but also systems of nonsmooth variational problems will be treated. As shown, the notion of sub and supersolution for nonlinear elliptic variational equations is an almost direct extension of the classic notion of sub and supersolution for the Laplace equation. A similar statement can be made for parabolic variational equations, for which the notion of sub and supersolution is a natural extension of the one for the heat equation. The situation is, however, diﬀerent for variational and hemivariational inequalities. Because of the intrinsic asymmetry of these problems (where the problems are stated as inequalities rather than as equalities), it is much more diﬃcult to deﬁne sub and supersolutions for variational and hemivariational inequalities. As an indispensable requirement, this notion should be an extension of the wellknown notion of sub and supersolution for variational equations. It seems to be the main reason that this powerful method and the comparison principles related with it have not been employed so far to investigate nonsmooth variational problems. The rapid development of the theory of variational and hemivariational inequalities and the proliﬁc growth of its numerous applications (see [124, 177]) made evident to us the need for a detailed and systematic exposition of the subsupersolution method for nonsmooth variational problems that covers the one for variational equations in a natural way. We have made eﬀorts to deﬁne a notion of sub and supersolution in such a way that will allow us to establish comparison principles for nonsmooth variational problems similar to the corresponding concepts for variational equations. The comparison principles based on the new notion of sub and supersolution will be seen to preserve many characteristic features of the elementary example on the real line considered above; i.e., we will be able to prove not only existence and enclosure of solutions for nonsmooth variational problems but also qualitative properties of the solution set, such as compactness and existence of smallest and greatest solutions. In addition, these new comparison principles will be shown to provide eﬀective tools to study noncoercive nonsmooth variational problems and permit more ﬂexible requirements on the growth rates of certain nonlinear data involved. This book is basically an outgrowth of the authors’ research on the subject during the past 10 years. It consists of seven chapters, including the introductory chapter. Each chapter begins with a short overview, and notes and remarks are added at the end. Chapter 2 provides needed mathematical prerequisites to make the book selfcontained. Chapter 3 deals with the subsupersolution method for weak solutions of nonlinear elliptic and parabolic variational equations, and it may be considered in some sense as a preparatory chapter to get to know some methods and techniques used also in later chapters. Chapter 4 to Chapter 7 form the core of the book dealing with 10 1 Introduction nonsmooth variational problems. Chapter 4 deals with multivalued elliptic and parabolic problems that involve multifunctions of Clarke’s subgradient type. The key notion of subsupersolution for variational inequalities is developed in Chapter 5. In Chapter 6, we deal with comparison principles for hemivariational inequalities and reveal their connection with the multivalued problems considered in Chapter 4. Finally, in Chapter 7, we treat variational– hemivariational inequalities and related problems such as eigenvalue problems for this kind of variational problems. Some important features of the monograph are as follows: • Presenting a systematic and uniﬁed exposition of the subsupersolution method for nonsmooth stationary and evolutionary variational problems, including variational and hemivariational inequalities. • Proving existence and comparison results, and characterizing the solution set topologically and order theoretically. • Inclusion of numerous new results, some of which have never been published. • Eﬀorts have been made to make the presentation selfcontained by providing the necessary mathematical background and theories in an extra chapter. • Attempts to draw a broad audience by writing the ﬁrst section of each chapter in a manner that emphasizes simple cases and ideas more than complicated reﬁnements. • Being accessible to graduate students in mathematics and engineering. • The power of the developed methodology is demonstrated through various examples and applications. 2 Mathematical Preliminaries In this chapter, we provide the mathematical background as it will be used in later chapters. 2.1 Basic Functional Analysis The purpose of this section is to provide a survey of basic results from functional analysis that will be used in the sequel. However, we will assume that the reader is familiar with some elementary notions such as metric spaces, Banach spaces, and Hilbert spaces, as well as notions related with the topological structure of these spaces. Unless otherwise indicated, all linear spaces considered in this book are assumed to be deﬁned over the real number ﬁeld R. The proofs of the results presented in this section can be found in standard textbooks, e.g., [5, 13, 24, 129, 200, 222]. 2.1.1 Operators in Normed Linear Spaces Let (X, · X ) and (Y, · Y ) be normed linear spaces, and let A : D(A) ⊂ X → Y be an operator with domain D(A) and range denoted by range(A). When D(A) = X, we write A : X → Y. Note that usually we drop the subscripts X and Y in the notation of the norms · X and · Y , respectively, if no ambiguity exists. Deﬁnition 2.1. Let A : D(A) ⊂ X → Y. (i) A is continuous at the point u ∈ D(A) iﬀ for each sequence (un ) in D(A), un → u implies Aun → Au. 12 2 Mathematical Preliminaries The operator A : D(A) ⊂ X → Y is called continuous iﬀ it is continuous at each point u ∈ D(A). (ii) A is called compact iﬀ A is continuous, and A maps bounded sets into relatively compact sets. Note that one sometimes uses the notion completely continuous for compact. For compact operators, the following ﬁxedpoint theorem from Schauder holds. Theorem 2.2 (Schauder’s FixedPoint Theorem). Let X be a Banach space, and let A:M →M be a compact operator that maps a nonempty subset M of X into itself. Then A has a ﬁxed point provided M is bounded, closed, and convex. In ﬁnitedimensional normed linear spaces, Theorem 2.2 reduces to Brouwer’s ﬁxedpoint theorem. Corollary 2.3 (Brouwer’s FixedPoint Theorem). If the operator A:M →M is continuous, then A has a ﬁxed point provided M is a compact, convex, nonempty subset in a ﬁnitedimensional normed linear space. Let A : D(A) ⊂ X → Y be a linear operator, which means that the domain D(A) of the operator A is a linear subspace of X and A satisﬁes A(αu + βv) = αAu + βAv for all u, v ∈ D(A), α, β ∈ R. Proposition 2.4. Let A : X → Y be a linear operator. Then the following two conditions are equivalent: (i) A is continuous. (ii) A is bounded; i.e., there is a constant c > 0 such that Au ≤ cu for all u ∈ X. For a linear continuous operator A : X → Y , the operator norm A is deﬁned by A = sup Au, u≤1 which can easily be shown to be equal to A = sup Au. u=1 2.1 Basic Functional Analysis 13 Proposition 2.5. Let L(X, Y ) denote the space of linear continuous operators A : X → Y, where X is a normed linear space and Y is a Banach space. Then L(X, Y ) is a Banach space with respect to the operator norm. Deﬁnition 2.6. Let A : D(A) ⊂ X → Y be a linear operator. The graph of A denoted by Gr(A) is deﬁned by the subset Gr(A) = {(u, Au) : u ∈ D(A)} of the product space X × Y. The operator A is called closed (or graphclosed) iﬀ Gr(A) is closed in X ×Y, which means that for each sequence (un ) in D(A), it follows from un → u in X and Aun → v in Y that u ∈ D(A) and v = Au. Finally, on D(A), the socalled graph norm · A is deﬁned by uA = u + Au for u ∈ D(A). Corollary 2.7. If X and Y are Banach spaces and A : D(A) ⊂ X → Y is closed, then D(A) equipped with the graph norm, i.e., (D(A), · A ), is a Banach space. Theorem 2.8 (Banach’s Closed Graph Theorem). Let X and Y be Banach spaces. Then each closed linear operator A : X → Y is continuous. For completeness, we shall recall the Uniform Boundedness Theorem and the Open Mapping Theorem, which together with Banach’s Closed Graph Theorem are all consequences of Baire’s Theorem. Theorem 2.9 (Uniform Boundedness Theorem). Let F be a nonempty set of continuous maps F : X → Y, where X is a Banach space and Y is a normed linear space. Assume that sup F u < ∞ F ∈F for each u ∈ X. Then a closed ball B in X of positive radius exists such that sup ( sup F u) < ∞. u∈B F ∈F 14 2 Mathematical Preliminaries Corollary 2.10 (Banach–Steinhaus Theorem). Let L ⊂ L(X, Y ) be a nonempty set of linear continuous operators A : X → Y, where X is a Banach space and Y is a normed linear space. Assume that sup Au < ∞ A∈L for each u ∈ X. Then supA∈L A < ∞. Theorem 2.11 (Banach’s Open Mapping Theorem). Let X and Y be Banach spaces and A : X → Y be a linear continuous operator. Then the following two conditions are equivalent: (i) A is surjective. (ii) A is open, which means that A maps open sets onto open sets. Corollary 2.12 (Banach’s Continuous Inverse Theorem). Let X and Y be Banach spaces and A : X → Y be a linear continuous operator. If the inverse operator A−1 : Y → X exists, then A−1 is continuous. Deﬁnition 2.13 (Embedding Operator). Let X and Y be normed linear spaces with X ⊂ Y. The embedding operator i : X → Y is deﬁned by i(u) = u; i.e., i is the identity operator from X into Y. (i) The embedding X ⊂ Y is called continuous iﬀ the embedding operator i : X → Y is continuous; i.e., a constant c > 0 exists such that uY ≤ c uX for all u ∈ X, which is equivalent with un → u in X implies un → u in Y. (ii) The embedding X ⊂ Y is called compact iﬀ the embedding operator i : X → Y is compact; i.e., i is continuous and each bounded sequence (un ) in X has a subsequence that converges in Y. Remark 2.14. More generally, one can deﬁne a continuous embedding of a normed linear space X into a normed linear space Y , whenever a linear, continuous, and injective operator i : X → Y exists. Similarly, X is compactly embedded into Y iﬀ a linear, compact, and injective operator i : X → Y exists. 2.1 Basic Functional Analysis 15 2.1.2 Duality in Banach Spaces Deﬁnition 2.15. Let X be a normed linear space. A linear continuous functional on X is a linear continuous operator f : X → R. The set of all linear continuous functionals on X is called the dual space X ∗ of X; i.e., X ∗ = L(X, R). For the image f (u) of the functional f at u ∈ X, we write f, u = f (u) u ∈ X, f ∈ X ∗ , and ·, · is called the duality pairing. According to the operator norm deﬁned in Sect. 2.1.1, the norm of f is given through f = sup  f, u . u≤1 As a consequence of Proposition 2.5, we get the following result. Corollary 2.16. Let X be a normed linear space. Then the dual space X ∗ is a Banach space with respect to the norm f for f ∈ X ∗ . The most important theorem about the structure of linear functionals on normed linear spaces is the Hahn–Banach Theorem. For real linear spaces, the Hahn–Banach Theorem reads as follows (see [24]). Theorem 2.17 (Hahn–Banach Theorem). Let p : E → R be a function on a real linear space E satisfying p(λx) = λp(x), ∀ x ∈ E, ∀ λ ≥ 0, p(x + y) ≤ p(x) + p(y), ∀ x, y ∈ E. Let G be a linear subspace of E, and let g : G → R be a linear functional such that g(x) ≤ p(x), ∀ x ∈ G. Then a linear functional f : E → R exists with the properties f (x) = g(x), ∀ x∈G f (x) ≤ p(x), ∀ x ∈ E. and As an immediate consequence from Theorem 2.17, we obtain the following theorem, which is the Hahn–Banach Theorem for normed linear spaces. 16 2 Mathematical Preliminaries Theorem 2.18. Let X be a normed linear space. Assume M is a linear subspace of X, and F : M → R is a linear functional such that F (u) ≤ c u for all u ∈ M, where c is some positive constant. Then F can be extended to a linear continuous functional f : X → R that satisﬁes  f, u  ≤ c u for all u ∈ X. First consequences from the Hahn–Banach Theorem are given in the following corollary. Corollary 2.19. Let X be a normed linear space. (i) For each given u0 ∈ X with u0 = 0, a functional f ∈ X ∗ exists such that f, u0 = u0 and f = 1. (ii) For all u ∈ X, one has u = sup f ∈X ∗ , f ≤1  f, u . (iii) If for u ∈ X the condition f, u = 0 for all f ∈ X ∗ holds, then u = 0. We set X ∗∗ = (X ∗ )∗ , which is called the bidual space and which consists of all linear continuous functionals F : X ∗ → R. Proposition 2.20. Let X be a normed linear space. The operator j : X → X ∗∗ deﬁned by j(u)(f ) = f, u for all u ∈ X, f ∈ X ∗ has the following properties: (i) j is linear and j(u) = u for all u ∈ X. (ii) j(X) is a closed subspace of X ∗∗ if and only if X is a Banach space. The operator j : X → X ∗∗ is called the canonical embedding of X into X ∗∗ . Deﬁnition 2.21. A normed linear space X is called reﬂexive if the canonical embedding j : X → X ∗∗ is surjective; i.e., j(X) = X ∗∗ . 2.1 Basic Functional Analysis 17 We readily observe that every reﬂexive normed linear space X is in fact a Banach space, which is isometrically isomorphic to X ∗∗ , and thus, we may write X = X ∗∗ . Corollary 2.22. (i) Each Hilbert space is reﬂexive. (ii) Every closed linear subspace of a reﬂexive Banach space X is again reﬂexive. (iii) The product of a ﬁnite number of reﬂexive Banach spaces is a reﬂexive Banach space. (iv) Let X and Y be two isomorphic normed linear spaces. If X is a reﬂexive Banach space, then Y is also a reﬂexive Banach space. (v) Let X be a Banach space. Then X is reﬂexive if and only if X ∗ is reﬂexive. (vi) If X is a separable and reﬂexive Banach space, then X ∗ is separable. Next we deﬁne the dual or adjoint operator of a linear operator A : D(A) ⊂ X → Y, where X and Y are two Banach spaces. Deﬁnition 2.23. Assume D(A) is dense in X. Then the dual operator A∗ : D(A∗ ) ⊂ Y ∗ → X ∗ is deﬁned by the following relation: A∗ v, u = v, Au for all v ∈ D(A∗ ), u ∈ D(A), where v ∈ Y ∗ belongs to D(A∗ ) if and only if a w ∈ X ∗ exists such that w, u = v, Au for all u ∈ D(A). To verify that A∗ is well deﬁned, we note ﬁrst that according to Deﬁnition 2.23, an element v ∈ Y ∗ belongs to D(A∗ ) if and only if a w ∈ X ∗ exists such that w, u = v, Au for all u ∈ D(A). We set A∗ v = w. As D(A) is dense in X, the element w is uniquely determined by v, and thus, the operator A∗ is well deﬁned. Moreover, one readily observes that A∗ is linear and graphclosed. In the special case that D(A) = X, we have the following results. Proposition 2.24. Let X and Y be two Banach spaces, and let A : X → Y be a linear and continuous operator. Then the dual operator A∗ : Y ∗ → X ∗ is also linear and continuous, and we have A∗ = A. Moreover, if the linear operator A : X → Y is compact, then so is the dual operator A∗ : Y ∗ → X ∗ . 18 2 Mathematical Preliminaries The following facts about the duality of embeddings are important, e.g., for the understanding of the concept of evolution triple, which will be introduced in Sect. 2.4.3. Proposition 2.25. Let X and Y be Banach spaces with X ⊂ Y such that X is dense in Y , and the embedding i:X→Y is continuous. Then the following is true: (i) The embedding Y ∗ ⊂ X ∗ is continuous, and the embedding operator î : Y ∗ → X ∗ is identical with the dual operator of i; i.e., î = i∗ . (ii) If X is, in addition, reﬂexive, then Y ∗ is dense in X ∗ . (iii) If the embedding X ⊂ Y is compact, then so is the embedding Y ∗ ⊂ X ∗ . Proof: As for (i), density arguments show that each element of Y ∗ can be uniquely identiﬁed with an element of X ∗ , and the continuity of the embedding Y ∗ ⊂ X ∗ follows from the continuity of i. The proof of (ii) makes use of the Hahn–Banach Theorem in connection with the reﬂexivity of X. (see [222, Chap. 18, 21]), and (iii) follows from Proposition 2.24. In ﬁnitedimensional Banach spaces, closed and bounded sets are compact. This result is no longer true for inﬁnitedimensional Banach spaces because of the following famous theorem due to Riesz. Theorem 2.26 (Riesz’ Lemma). Let X be a normed linear space. Then, the closed unit ball in X is compact if and only if X is ﬁnitedimensional. According to Theorem 2.26, in inﬁnitedimensional Banach spaces, there are bounded sequences that have no convergent subsequence. This lack of compactness in inﬁnitedimensional spaces is one of the main reasons for many diﬃculties in the functional analytical treatment of variational problems. To overcome these diﬃculties, new concepts of convergence (or new topologies) have been introduced with respect to which the unit ball is compact (respectively, sequentially compact). Deﬁnition 2.27. Let X be a Banach space. A sequence (un ) ⊂ X is called weakly convergent in X to an element u ∈ X iﬀ f, un → f, u for all f ∈ X ∗ . The weak convergence is denoted by un u as n → ∞ or w− lim un = u. n→∞ Note, in contrast to the weak convergence, we call the usual convergence with respect to the norm (un → u) sometimes the strong convergence. The following theorem provides a compactness result with respect to the topology introduced by the weak convergence. 2.1 Basic Functional Analysis 19 Theorem 2.28 (Eberlein–Smulian Theorem). Let X be a reﬂexive Banach space. Then, each bounded sequence (un ) ⊂ X has a weakly convergent subsequence. A few properties of weak convergence are summarized in the next proposition. Proposition 2.29. Let X be Banach spaces, and (un ) ⊂ X. (i) un → u implies un u. (ii) If X is ﬁnitedimensional, then strong and weak convergence are equivalent. (iii) If un u, then (un ) is bounded and u ≤ lim inf un . n→∞ (iv) If un ∗ u in X and fn → f in X , then it follows that fn , un → f, u . (v) If un → u in X and fn f in X ∗ , then it follows that fn , un → f, u . The reverse of the Eberlein–Smulian Theorem is also true; i.e, a Banach space is reﬂexive if and only if every bounded sequence has a weakly convergent subsequence. Thus, the compactness result given by Theorem 2.28 is only valid in reﬂexive Banach spaces. To deal with nonreﬂexive Banach spaces, the following socalled weak∗ convergence has been introduced. Deﬁnition 2.30. Let X be a Banach space. A sequence (fn ) ⊂ X ∗ is called weakly∗ convergent to an element f ∈ X ∗ iﬀ fn , u → f, u for all u ∈ X. The weak∗ convergence is denoted by fn ∗ f as n → ∞, or w∗ − lim fn = f. n→∞ Proposition 2.31. Let X be a Banach space, and let (fn ) be a sequence in the dual space X ∗ . (i) fn → f in X ∗ implies fn ∗ f. (ii) If fn ∗ f, then (fn ) is bounded in X ∗ and f ≤ lim inf fn . n→∞ (iii) If un → u in X and fn ∗ f in X ∗ , then it follows that fn , un → f, u . 20 2 Mathematical Preliminaries (iv) fn f in X ∗ implies fn (v) If X is reﬂexive, then fn ∗ ∗ f. f is equivalent to fn f. Deﬁnition 2.32. Let A : X → Y be a linear operator, where X and Y are Banach spaces. A is called weakly sequentially continuous iﬀ un u implies Aun Au. A is called strongly continuous iﬀ u un implies Aun → Au. A few simple consequences are provided in the next proposition. Proposition 2.33. Let A : X → Y be a linear operator, where X and Y are Banach spaces. (i) If A is continuous, then A is weakly sequentially continuous. (ii) If A is compact, then A is strongly continuous. (iii) If A is strongly continuous and X is reﬂexive, then A is compact. 2.1.3 Convex Analysis and Calculus in Banach Spaces Let X be a normed linear space. A subset K of X is convex iﬀ u, v ∈ K implies tu + (1 − t)v ∈ K for all 0 ≤ t ≤ 1. Theorem 2.34. Let H be a Hilbert space with inner product (·, ·), and let K be a nonempty, closed, and convex subset of H. Then to each u ∈ H, a uniquely deﬁned v ∈ K closest to u exists, that is, v∈K: u − v = inf u − w. w∈K Equivalently, v ∈ K is the uniquely deﬁned solution of the variational inequality v ∈ K : (u − v, w − v) ≤ 0 for all w ∈ K. Consequences of Theorem 2.34 are the wellknown Orthogonal Projection Theorem and the Riesz Representation Theorem of linear continuous functionals on Hilbert spaces. The latter implies that a Hilbert space H is isometrically isomorphic with its dual space H ∗ . A generalization of the Riesz Representation Theorem is the Lax–Milgram Theorem (see Sect. 2.3). Important consequences of the Hahn–Banach Theorem are various separation theorems, such as the following one. Theorem 2.35 (Separation Theorem). Let X be a normed linear space, and let K ⊂ X be a closed and convex subset. If u0 ∈ X \ K, then a linear continuous functional f ∈ X ∗ and an α ∈ R exists such that f, u ≤ α for all u ∈ K, and f, u0 > α. 2.1 Basic Functional Analysis 21 Deﬁnition 2.36. A subset M of a normed linear space X is called weakly sequentially closed if the limit of every weakly convergent sequence (un ) ⊂ M belongs to M ; i.e., (un ) ⊂ M and un u implies u ∈ M. Simple examples show that, in general, closed sets of a normed linear space need not be weakly sequentially closed. However, by means of Theorem 2.35, one gets the following equivalence. Proposition 2.37. Let M be a convex subset of a normed linear space X. Then, M is closed if and only if M is weakly sequentially closed. Next we present some convexity and smoothness properties of the norm in Banach spaces that are important for proving existence results for abstract operator equations involving operators of monotone type (see Theorem 2.156 in Sect. 2.4.4). Deﬁnition 2.38. A Banach space X is called strictly convex if and only if tu + (1 − t)v < 1 provided that u = v = 1, u = v, and 0 < t < 1. A Banach space X is called locally uniformly convex if and only if for each ε ∈ (0, 2], and for each u ∈ X with u = 1, a δ(ε, u) > 0 exists such that for all v with v = 1 and u − v ≥ ε, the following holds: 1 u + v ≤ 1 − δ(ε, u). 2 A Banach space X is called uniformly convex if and only if X is locally uniformly convex and δ can be chosen to be independent of u. Obviously we have the following implications: uniformly convex =⇒ locally uniformly convex =⇒ strictly convex. Example 2.39. Each Hilbert space is uniformly convex. This readily follows from the parallelogram identity 2 2 1 (u − v) + 1 (u + v) = 1 (u2 + v2 ). 2 2 2 Example 2.40. Let 1 < p < ∞ and Ω ⊂ RN be a domain; then from Clarkson’s inequality (see Sect. 2.2.4), it follows that Lp (Ω) is uniformly convex. By using this result, one readily sees that the Sobolev spaces W m,p (Ω) are uniformly convex too, for 1 < p < ∞ and m = 0, 1, . . . . Furthermore, the following theorems hold. 22 2 Mathematical Preliminaries Theorem 2.41 (Milman–Pettis Theorem). Each uniformly convex Banach space is reﬂexive. Convexity properties of the norm are closely related with smoothness properties of the norm, i.e., the smoothness of the function u → u. Theorem 2.42. Let X be a reﬂexive Banach space. Then the following holds: (i) If X ∗ is strictly convex, then the function u → u is Gâteauxdiﬀerentiable on X \ {0}. (ii) If X ∗ is locally uniformly convex, then the function u → u is Fréchetdiﬀerentiable on X \ {0}. (iii) (Troyanski) In every reﬂexive Banach space X, an equivalent norm can be introduced so that both X and X ∗ are locally uniformly convex. The notions of Gâteaux and Fréchet derivatives that occur in Theorem 2.42 are natural generalizations of the directional and total derivative of functions f : Rn → Rm , respectively, to mappings between Banach spaces. In particular, in the calculus of variations, these notions allow us to generalize the classic criteria in the study of extrema for realvalued functions in Rn to realvalued functionals F : D(F ) ⊂ X → R deﬁned on a subset of a Banach space X. Deﬁnition 2.43 (Gâteaux Derivative). Let X and Y be Banach spaces, and let f : U ⊂ X → Y be a map whose domain D(f ) = U is an open subset of X. The directional derivative of f at u ∈ U in the direction h ∈ X is given by f (u + th) − f (u) δf (u; h) = lim t→0 t provided this limit exists. If δf (u; h) exists for every h ∈ X, and if the mapping DG f (u) : X → Y deﬁned by DG f (u)h = δf (u; h) is linear and continuous, then we say that f is Gâteauxdiﬀerentiable at u, and we call DG f (u) the Gâteaux derivative of f at u. Deﬁnition 2.44 (Fréchet Derivative). Let X and Y be Banach spaces, and let f : U ⊂ X → Y, where the domain D(f ) = U is an open subset of X. Then f is called Fréchetdiﬀerentiable at u ∈ U if and only if a linear and continuous mapping A : X → Y exists such that lim h→0 f (u + h) − f (u) − Ah =0 h or equivalently f (u + h) − f (u) = Ah + o(h), (h → 0). If such a mapping A exists, then we call DF f (u) = A (or simply f (u) = A) the Fréchet derivative of f at u. 2.1 Basic Functional Analysis 23 Corollary 2.45. Let X and Y be Banach spaces, and let f : U ⊂ X → Y. Then the following relations between Gâteaux and Fréchet derivative hold: (i) If f is Fréchetdiﬀerentiable at u ∈ U, then f is Gâteauxdiﬀerentiable at u. (ii) If f is Gâteauxdiﬀerentiable in a neighborhood of u0 and DG f is continuous at u0 , then f is Fréchetdiﬀerentiable at u0 and f (u0 ) = DG f (u0 ). Remark 2.46. If f : U ⊂ X → Y is Fréchetdiﬀerentiable in U and f : U → L(X, Y ) is continuous, then we write f ∈ C 1 (U ; Y ) or simply f ∈ C 1 (U ) if Y = R. In a similar way as for mappings from Rn into Rm , one can prove chain rules for both the Fréchet and the Gâteaux derivative. Example 2.47. Let X = Lp (Ω), where 1 < p < ∞. We will compute the Gâteaux derivative of the p th power Lp norm, i.e., of the function f : X → R deﬁned by f (u) = upLp (Ω) . After elementary calculations, we get DG f (u)h = δf (u; h) = p Ω up−2 uh dx if we consider realvalued functions u : Ω → R. In case the functions are complexvalued, we get p δf (u; h) = up−2 (ūh + uh̄) dx. 2 Ω We introduce next the notions of convex and semicontinuous functions (or functionals). Deﬁnition 2.48 (Semicontinuous, Convex Functionals). Let X be a Banach space and φ : M ⊂ X → [−∞, ∞] with M = D(φ). (i) The functional φ is called sequentially lower semicontinuous at u ∈ M if and only if φ(u) ≤ lim inf φ(un ) n→∞ (2.1) holds for each sequence (un ) ⊂ M such that un → u as n → ∞. (ii) The functional φ is called lower semicontinuous if and only if the set Mr is closed relative to M for all r ∈ R, where Mr = {u ∈ M : φ(u) ≤ r}. (iii) The functional φ is called weak sequentially lower semicontinuous at u ∈ M if and only if (2.1) holds for each weakly convergent sequence (un ) to u. u, i.e., un 24 2 Mathematical Preliminaries (iv) The functional φ is called sequentially upper semicontinuous (respectively, weak sequentially upper semicontinuous, upper semicontinuous) if and only if −φ is sequentially lower semicontinuous (respectively, weak sequentially lower semicontinuous, lower semicontinuous). (v) The functional φ is called convex if and only if M is convex and φ(tu + (1 − t)v) ≤ tφ(u) + (1 − t)φ(v), 0 ≤ t ≤ 1, (2.2) for all u, v ∈ M for which the righthand side of (2.2) is meaningful; φ is called strictly convex if and only if for all t with 0 < t < 1 and for all u, v ∈ M with u = v inequality (2.2) holds strictly; i.e., (2.2) holds with ≤ replaced by <. The following proposition provides the connection between the above notions. Proposition 2.49. Let X be a Banach space and φ : M ⊂ X → [−∞, ∞] with M = D(φ). (i) φ is sequentially lower semicontinuous on M if and only if φ is lower semicontinuous on M. (ii) Assume u ∈ M with φ(u) = ±∞. Then φ is sequentially lower semicontinuous at u if and only if, for each ε > 0, a δ(ε) > 0 exists such that for all v ∈ M with v − u < δ(ε) implies φ(u) < φ(v) + ε. (iii) φ is continuous if and only if φ is both lower and upper semicontinuous. (iv) If, in addition, M is closed and convex, and φ is convex, then lower semicontinuous, sequentially lower semicontinuous and weak sequentially lower semicontinuous are mutually equivalent. Let X be a Banach space. In what follows we consider only convex functionals φ : X → R ∪ {+∞}; i.e., we do not allow “−∞” as a value for the convex functional φ. The reason is that if φ(u0 ) = −∞ at some point u0 and if, in addition, φ is lower semicontinuous, then φ would be nowhere ﬁnite. This can readily be seen by the following arguments. Assume there is some u ∈ X with φ(u) ∈ R. Then from the convexity we get for all t ∈ (0, 1), φ(tu0 + (1 − t)u) = −∞. Taking the limit t → 0, the lower semicontinuity yields φ(u) = −∞, a contradiction. Deﬁnition 2.50. Let X be a Banach space and φ : X → R ∪ {+∞} be a convex functional. (i) The eﬀective domain of φ is the set dom(φ) deﬁned by dom(φ) = {u ∈ X : φ(u) < +∞}. (ii) φ is said to be proper if dom(φ) = ∅. 2.1 Basic Functional Analysis 25 (iii) The epigraph of φ, denoted by epi(φ), is given by epi(φ) = {(u, λ) ∈ X × R : φ(u) ≤ λ}. We summarize some elementary properties of convex functionals as follows. Corollary 2.51. Let X be a Banach space, and let φ, φi : X → R ∪ {+∞}, i = 1, 2, be convex functionals. Then the following holds: (i) (ii) (iii) (iv) dom(φ) is convex. If λ ≥ 0, then λφ is convex. If φ1 and φ2 are convex, then φ1 + φ2 is convex. φ is convex, proper, and lower semicontinuous if and only if epi(φ) is, respectively, convex, nonempty, and closed in X × R. Proposition 2.52. Let X be a Banach space, and let φ : X → R ∪ {+∞} be a convex, proper, and lower semicontinuous functional. Then φ is locally Lipschitz on the interior of dom(φ). Theorem 2.53 (Weierstrass’ Theorem). Let X be a reﬂexive Banach space. If φ : X → R ∪ {+∞} is a convex, proper, and lower semicontinuous functional satisfying lim φ(u) = +∞, u→∞ then the problem u∈X: φ(u) = inf φ(v) v∈X admits at least one solution. The following notion of subgradient generalizes the classic concept of a derivative. Deﬁnition 2.54 (Subdiﬀerential). Let X be a Banach space, and let φ : X → R ∪ {+∞} be a convex and proper functional. An element u∗ ∈ X ∗ is called a subgradient of φ at u ∈ dom(φ) if and only if the following inequality holds: φ(v) ≥ φ(u) + u∗ , v − u for all v ∈ X. (2.3) The set of all u∗ ∈ X ∗ satisfying (2.3) is called the subdiﬀerential of φ at u ∈ dom(φ), and is denoted by ∂φ(u). First properties of the subdiﬀerential are given in the following proposition. Proposition 2.55. Let X be a Banach space, and let φ : X → R ∪ {+∞} be a convex and proper functional. Then we have the following properties of ∂φ: 26 2 Mathematical Preliminaries (i) ∂φ(u) is convex and weak∗ closed. (ii) If φ is continuous at u ∈ dom(φ), then ∂φ(u) is nonempty, convex, bounded, and weak∗ compact. Note, in (i) of Proposition 2.55 ∂φ(u) = ∅ is possible. Proposition 2.56. Let X be a Banach space, and let φ : X → R ∪ {+∞} be a convex and proper functional. If φ is Gâteauxdiﬀerentiable at u ∈ int(dom(φ)), then ∂φ(u) = {DG φ(u)}. If φ is continuous at u and ∂φ(u) is a singleton, then φ is Gâteauxdiﬀerentiable at u. The following sum rule for the subdiﬀerential is due to Moreau and Rockafellar. Proposition 2.57 (Sum Rule). Let X be a Banach space, and let φ1 , φ2 : X → R ∪ {+∞} be convex functionals. If there is a point u0 ∈ dom(φ1 ) ∩ dom(φ2 ) at which φ1 is continuous, then the following holds: ∂(φ1 + φ2 )(u) = ∂φ1 (u) + ∂φ2 (u) for all u ∈ X. Example 2.58. Let f : R → R be a nondecreasing function with its onesided limits f and f¯. Deﬁne φ : R → R by x φ(x) = x f (s) ds = x0 f¯(s) ds. x0 Note that φ is convex and ﬁnite on R, i.e., dom(φ) = R, and thus φ is even locally Lipschitz. Elementary calculations show that the subdiﬀerential is given by ∂φ(x) = [f (x), f¯(x)]. Example 2.59. Let φ : R → R ∪ {+∞} be a convex, proper, lower semicontinuous function, and Ω ⊂ RN a Lebesguemeasurable set such that either 0 = φ(0) = mins∈R φ(s) or the measurable set Ω has ﬁnite measure. Deﬁne Φ : Lp (Ω) → R ∪ {+∞}, 1 < p < ∞, by Φ(u) = φ(u(x)) dx if φ(u) ∈ L1 (Ω), +∞ otherwise. Ω p Then Φ : L (Ω) → R ∪ {+∞} is convex, proper, lower semicontinuous, and u∗ ∈ ∂Φ(u) if and only if u∗ ∈ Lq (Ω), and u∗ (x) ∈ ∂φ(u(x)), for a.e. x ∈ Ω, where q is the Hölder conjugate; i.e., 1/p + 1/q = 1. 2.1 Basic Functional Analysis 27 2.1.4 Partially Ordered Sets Deﬁnition 2.60 (Partially Ordered Set). Let P be a nonempty set. We say that a relation x ≤ y between certain pairs of elements of P is a partial ordering in P , and that (P, ≤) is a partially ordered set, if “≤” has the following properties: (i) x ≤ x for all x ∈ P (reﬂexivity). (ii) If x ≤ y and y ≤ x, then x = y (antisymmetry). (iii) If x ≤ y and y ≤ z, then x ≤ z (transitivity). Note that x < y stands for x ≤ y and x = y. Next we deﬁne several notions based on the partial ordering introduced above. Deﬁnition 2.61. Let (P, ≤) be a partially ordered set. (i) (ii) (iii) (iv) (v) (vi) (vii) An element b of P is called an upper bound of a subset A of P if x ≤ b for each x ∈ A. If b ∈ A, we say that b is the greatest element of A. A lower bound of A and the smallest element of A are deﬁned similarly, replacing x ≤ b above by b ≤ x. If the set of all upper bounds of A has the minimum, we call it a least upper bound of A and denote it by sup A. The greatest lower bound, inf A, of A is deﬁned similarly. An element x ∈ A is called a maximal element of A ⊂ P, if there is no y = x in A for which x ≤ y. Similarly, a minimal element of A is deﬁned. Obviously, every greatest element of A is a maximal element of A. We say that a partially ordered set P is a lattice if inf{x, y} and sup{x, y} exist for all x, y ∈ P . A subset C of P is said to be upward directed if for each pair x, y ∈ C there is a z ∈ C such that x ≤ z and y ≤ z, and C is downward directed if for each pair x, y ∈ C there is a w ∈ C such that w ≤ x and w ≤ y. If C is both upward and downward directed, it is called directed. A subset C of a partially ordered set P is called a chain if x ≤ y or y ≤ x for all x, y ∈ C. We say that C is well ordered if each nonempty subset of C has a minimum, and inversely well ordered if each nonempty subset of C has a maximum. Obviously, each (inversely) wellordered set is a chain and each chain is directed. Theorem 2.62 (Zorn’s Lemma). If in a partially ordered set P, every chain has an upper bound, then P possesses a maximal element. 28 2 Mathematical Preliminaries 2.2 Sobolev Spaces In this section, we summarize the main properties of Sobolev spaces. These properties include, e.g., the approximation of Sobolev functions by smooth functions (density theorems), continuity properties and compactness conditions (embedding theorems), the deﬁnition of the boundary values of Sobolev functions (trace theorem), and calculus for Sobolev functions (chain rule). 2.2.1 Spaces of Lebesgue Integrable Functions Let RN , N ≥ 1, be equipped with the Lebesgue measure, and let Ω ⊂ RN be a domain; i.e., Ω is an open and connected subset of RN . For 1 ≤ p < ∞, we denote by Lp (Ω) the Banach space of measurable functions u : Ω → R with respect to the norm uLp (Ω) = Ω p 1/p u dx < ∞. For a measurable function u, we put uL∞ (Ω) = inf{α ∈ R : meas ({x ∈ Ω : u(x) > α}) = 0}. We denote by L∞ (Ω) the Banach space of all measurable functions f satisfying uL∞ (Ω) < ∞. We also introduce the local Lp spaces, denoted by Lploc (Ω). A function u belongs to Lploc (Ω) if it is measurable and up dx < ∞ K for every compact subset K of Ω. The following main theorems can be found in standard textbooks on real analysis and measure theory (see [201, 114]). Theorem 2.63 (Lebesgue’s Dominated Convergence Theorem). Suppose (un ) is a sequence in L1 (Ω) such that u(x) = lim un (x) n→∞ exists almost everywhere (a.e.) on Ω. If there is a function g ∈ L1 (Ω) such that, for a.e. x ∈ Ω, and for all n = 1, 2, . . . , un (x) ≤ g(x) then u ∈ L1 (Ω) and lim n→∞ Ω un − u dx = 0. 2.2 Sobolev Spaces 29 In some sense the following reverse statement of Theorem 2.63 holds. Theorem 2.64. Let un , u ∈ L1 (Ω), n ∈ N, be such that lim un − u dx = 0. n→∞ Ω Then a subsequence (unk ) of (un ) exists with unk (x) → u(x) for a.e. x ∈ Ω. Theorem 2.65 (Fatou’s Lemma). Let (un ) be a sequence of measurable functions, and let g ∈ L1 (Ω). If un ≥ g then we have a.e. on Ω, lim inf un dx ≤ lim inf Ω n→∞ n→∞ Ω un dx. If Ω ⊂ RN is a measurable subset, we denote its Lebesgue measure by meas(Ω) = Ω. Theorem 2.66 (Egorov’s Theorem). Let (un ), u be measurable functions, and un → u a.e. on Ω, where Ω ⊂ RN is measurable with Ω < ∞. Then for each ε > 0, a measurable subset E ⊂ Ω exists such that (i) Ω \ E < ε. (ii) un → u uniformly on E. A characterization of the dual spaces of Lp (Ω) is given in the next theorem. Theorem 2.67 (Dual Space). Let Ω ⊂ RN be a domain, and let Φ be a linear continuous functional on Lp (Ω), 1 < p < ∞. Then a uniquely deﬁned function g ∈ Lq (Ω) exists with q satisfying 1/p + 1/q = 1 such that Φ, u = g u dx for all u ∈ Lp (Ω) Ω and Φ(Lp (Ω))∗ = gLq (Ω) . If Φ is a linear continuous functional on L1 (Ω), then a uniquely deﬁned function g ∈ L∞ (Ω) exists such that Φ, u = g u dx for all u ∈ L1 (Ω) Ω and Φ(L1 (Ω))∗ = gL∞ (Ω) . 30 2 Mathematical Preliminaries In view of Theorem 2.67, the dual space of Lp (Ω) is isometrically isomorphic to Lq (Ω) for 1 ≤ p < ∞ with q = ∞ if p = 1. We summarize some important properties of Lp spaces in the following theorem. Theorem 2.68. Let Ω ⊂ RN be a domain. (i) (ii) (iii) (iv) (v) For 1 ≤ p < ∞, the spaces Lp (Ω) are separable. L∞ (Ω) is not separable. For 1 < p < ∞, the spaces Lp (Ω) are reﬂexive. L1 (Ω) and L∞ (Ω) are not reﬂexive. For 1 < p < ∞, the spaces Lp (Ω) are uniformly convex. 2.2.2 Deﬁnition of Sobolev Spaces Let α = (α1 , . . . , αN ) with nonnegative integers α1 , . . . , αN be a multiindex, and denote its order by α = α1 + · · · + αN . Set Di = ∂/∂xi , i = 1, . . . , N, αN and Dα u = D1α1 · · · DN u, with D0 u = u. Let Ω be a domain in RN with 1 N ≥ 1. Then w ∈ Lloc (Ω) is called the αth weak or generalized derivative of u ∈ L1loc (Ω) if and only if α α uD ϕ dx = (−1) wϕ dx, for all ϕ ∈ C0∞ (Ω), Ω Ω C0∞ (Ω) holds, where denotes the space of inﬁnitely diﬀerentiable functions with compact support in Ω. The generalized derivative w denoted by w = Dα u is unique up to a change of the values of w on a set of Lebesgue measure zero. Deﬁnition 2.69. Let 1 ≤ p ≤ ∞ and m = 0, 1, 2, . . . . The Sobolev space W m,p (Ω) is the space of all functions u ∈ Lp (Ω), which have generalized derivatives up to order m such that Dα u ∈ Lp (Ω) for all α: α ≤ m. For m = 0, we set W 0,p (Ω) = Lp (Ω). With the corresponding norms given by ⎞1/p ⎛ uW m,p (Ω) = ⎝ α≤m Dα upLp (Ω) ⎠ , 1 ≤ p < ∞, uW m,∞ (Ω) = max Dα uL∞ (Ω) , α≤m W m,p (Ω) becomes a Banach space. Deﬁnition 2.70. W0m,p (Ω) is the closure of C0∞ (Ω) in W m,p (Ω). W0m,p (Ω) is a Banach space with the norm · W m,p (Ω) . Before we summarize some basic properties of Sobolev spaces, we need to classify the regularity of boundaries. 2.2 Sobolev Spaces 31 Deﬁnition 2.71. Let Ω ⊂ RN be a bounded domain, with boundary ∂Ω. We say that the boundary ∂Ω is of class C k,λ , k ∈ N0 , λ ∈ (0, 1], if there are m ∈ N Cartesian coordinate systems Cj , j = 1, . . . , m, Cj = (xj,1 , . . . , xj,N −1 , xj,N ) = (xj , xj,N ) and real numbers α, β > 0, as well as m functions aj with aj ∈ C k,λ ([−α, α]N −1 ), j = 1, . . . , m, such that the sets deﬁned by Λj = {(xj , xj,N ) ∈ RN : xj  ≤ α, xj,N = aj (xj )}, V+j = {(xj , xj,N ) ∈ RN : xj  ≤ α, aj (xj ) < xj,N < aj (xj ) + β}, V−j = {(xj , xj,N ) ∈ RN : xj  ≤ α, aj (xj ) − β < xj,N < aj (xj )}, possess the following properties: Λj ⊂ ∂Ω, V+j ⊂ Ω, V−j ⊂ RN \ Ω, j = 1, . . . , m, and m Λj = ∂Ω. j=1 Remark 2.72. If ∂Ω ∈ C 0,1 , then we call ∂Ω a Lipschitz boundary, which means that ∂Ω is locally the graph of a Lipschitz continuous function. In this case, the (N − 1)dimensional surface measure is well deﬁned, on the basis of which Lp (∂Ω)spaces can be introduced (see [66]). As Lipschitz continuous functions admit a.e. a gradient, the outer unit normal on ∂Ω exists for a.a. x ∈ ∂Ω (see [94]), which allows us to extend the integration by parts formula to Sobolev functions on Lipschitz domains. Theorem 2.73. Let Ω ⊂ RN be a bounded domain, N ≥ 1. Then we have the following: (i) W m,p (Ω) is separable for 1 ≤ p < ∞. (ii) W m,p (Ω) is reﬂexive for 1 < p < ∞. (iii) Let 1 ≤ p < ∞. Then C ∞ (Ω) ∩ W m,p (Ω) is dense in W m,p (Ω), and if ∂Ω is a Lipschitz boundary, then C ∞ (Ω) is dense in W m,p (Ω), where C ∞ (Ω) and C ∞ (Ω) are the spaces of inﬁnitely diﬀerentiable functions in Ω and Ω, respectively (cf. [99]). As for the proofs of these properties we refer to [99]. Now we state some Sobolev embedding theorems. Let X, Y be two normed linear spaces with X ⊂ Y . We recall the operator i : X → Y deﬁned by i(u) = u for all u ∈ X is called the embedding operator of X into Y . We say X is continuously (compactly) embedded in Y if X ⊂ Y and the embedding operator i : X → Y is continuous (compact). 32 2 Mathematical Preliminaries Theorem 2.74 (Sobolev Embedding Theorem). Let Ω ⊂ RN , N ≥ 1, be a bounded domain with Lipschitz boundary ∂Ω. Then the following holds: ∗ (i) If mp < N, then the space W m,p (Ω) is continuously embedded in Lp (Ω), p∗ = N p/(N − mp), and compactly embedded in Lq (Ω) for any q with 1 ≤ q < p∗ . (ii) If 0 ≤ k < m − Np < k + 1, then the space W m,p (Ω) is continuously em bedded in C k,λ (Ω), λ = m − Np − k, and compactly embedded in C k,λ (Ω) for any λ < λ. (iii) Let 1 ≤ p < ∞, then the embeddings Lp (Ω) ⊃ W 1,p (Ω) ⊃ W 2,p (Ω) ⊃ · · · are compact. Here C k,λ (Ω) denotes the Hölder space; cf. [99]. As for the proofs we refer to, e.g., [99, 222]. The proper deﬁnition of boundary values for Sobolev functions is based on the following theorem. Theorem 2.75 (Trace Theorem). Let Ω ⊂ RN be a bounded domain with Lipschitz (C 0,1 ) boundary ∂Ω, N ≥ 1, and 1 ≤ p < ∞. Then exactly one continuous linear operator exists γ : W 1,p (Ω) → Lp (∂Ω) such that: (i) γ(u) = u∂Ω if u ∈ C 1 (Ω). (ii) γ(u)Lp (∂Ω) ≤ C uW 1,p (Ω) with C depending only on p and Ω. (iii) If u ∈ W 1,p (Ω), then γ(u) = 0 in Lp (∂Ω) if and only if u ∈ W01,p (Ω). Deﬁnition 2.76 (Trace). We call γ(u) the trace (or generalized boundary function) of u on ∂Ω. Remark 2.77. We note that the trace operator γ : W 1,p (Ω) → Lp (∂Ω) in Theorem 2.75 is not surjective; i.e., there are functions ϕ ∈ Lp (∂Ω) that are not the traces of functions u from W 1,p (Ω). To describe precisely the range of the trace operator, Sobolev spaces of fractional order, usually referred to as Sobolev–Slobodeckij spaces, have to be taken into account (see [90, 132, 213, 219]). From [132, Theorem 6.8.13, Theorem 6.9.2], we obtain the following result. Theorem 2.78. Let Ω ⊂ RN be a bounded domain with Lipschitz boundary ∂Ω, N ≥ 1, and 1 < p < ∞. Then 1 γ(W 1,p (Ω)) = W 1− p ,p (∂Ω). 2.2 Sobolev Spaces 33 The following compactness result of the trace operator holds (see [132]). Theorem 2.79. Let Ω ⊂ RN be a bounded domain with Lipschitz boundary ∂Ω, N ≥ 1. (i) If 1 < p < N, then γ : W 1,p (Ω) → Lq (∂Ω) is completely continuous for any q with 1 ≤ q < (N p − p)/(N − p). (ii) If p ≥ N, then for any q ≥ 1, γ : W 1,p (Ω) → Lq (∂Ω) is completely continuous. Sobolev–Slobodeckij spaces form a scale of continuous and even compact embeddings with respect to their fractional order of regularity. More precisely, we can deduce the following compact embedding result for the spaces W l,2 (Ω) with l ∈ R+ from [219, Theorem 7.9, Theorem 7.10]. Theorem 2.80. Let Ω ⊂ RN be a bounded domain with Lipschitz boundary ∂Ω, N ≥ 1, and let l2 < l1 ≤ 1, where l1 , l2 ∈ R+ . Then the embedding W l1 ,2 (Ω) ⊂ W l2 ,2 (Ω) is compact. If M is a C k,κ manifold (C 0,1 stands for Lipschitzmanifold) and l2 < l1 < k + κ with l1 , l2 ∈ R+ (for l1 integer, l1 = k + κ is admissible), then the embedding W l1 ,2 (M ) ⊂ W l2 ,2 (M ) is compact. In a similar way as for Sobolev spaces we have the following trace theorem, which can be deduced from [219, Theorem 8.7]. Theorem 2.81 (Trace Theorem). Let Ω ⊂ RN be a bounded domain with Lipschitz boundary ∂Ω, N ≥ 1, and let 1/2 < l ≤ 1 with l ∈ R+ . Then a uniquely deﬁned continuous linear operator exists γ : W l,2 (Ω) → W l−1/2,2 (∂Ω) such that γ(u) = u∂Ω if u ∈ C 1 (Ω). Theorem 2.80 and Theorem 2.81 hold likewise in the general case of the spaces W l,p (Ω) with l ∈ R+ , 1 < p < ∞, and can be found, e.g., in [90, 132, 212, 213, 219]. The following extension result is useful in the study of unbounded domain problems. 34 2 Mathematical Preliminaries Lemma 2.82. Let Ω0 ⊂⊂ Ω, that is, Ω0 is compactly contained in Ω. Assume g ∈ W 1,p (Ω), u ∈ W 1,p (Ω0 ), and u − g ∈ W01,p (Ω0 ), 1 ≤ p < ∞. Then the function w deﬁned by w(x) = u(x) if g(x) if x ∈ Ω0 , x ∈ Ω \ Ω0 is in W 1,p (Ω), and its generalized derivative Di w = ∂w/∂xi , i = 1, . . . , N, is given by Di u(x) if x ∈ Ω0 , Di w(x) = Di g(x) if x ∈ Ω \ Ω0 . For the proof of Lemma 2.82, see [120, Lemma 20.14]. Its proof is based on the density property (iii) of Theorem 2.73 and the characterization of the traces of W01,p (Ω) function. 2.2.3 Chain Rule and Lattice Structure In this section, we assume that Ω ⊂ RN is a bounded domain with Lipschitz boundary ∂Ω. Lemma 2.83 (Chain Rule). Let f ∈ C 1 (R) and sups∈R f (s) < ∞. Let 1 ≤ p < ∞ and u ∈ W 1,p (Ω). Then the composite function f ◦ u ∈ W 1,p (Ω), and its generalized derivatives are given by Di (f ◦ u) = (f ◦ u)Di u, i = 1, . . . , N. Lemma 2.84 (Generalized Chain Rule). Let f : R → R be continuous and piecewise continuously diﬀerentiable with sups∈R f (s) < ∞, and u ∈ W 1,p (Ω), 1 ≤ p < ∞. Then f ◦ u ∈ W 1,p (Ω), and its generalized derivative is given by Di (f ◦ u)(x) = f (u(x))Di u(x) 0 if f is diﬀerentiable at u(x) , otherwise. The chain rule may further be extended to Lipschitz continuous f ; see [99, 222]. Lemma 2.85 (Generalized Chain Rule). Let f : R → R be a Lipschitz continuous function and u ∈ W 1,p (Ω), 1 ≤ p < ∞. Then f ◦ u ∈ W 1,p (Ω), and its generalized derivative is given by Di (f ◦ u)(x) = fB (u(x))Di u(x) for a.e. x ∈ Ω, where fB : R → R is a Borelmeasurable function such that fB = f a.e. in R. 2.2 Sobolev Spaces 35 The generalized derivative of the following special functions are frequently used in later chapters. Example 2.86. Let 1 ≤ p < ∞ and u ∈ W 1,p (Ω). Then u+ = max{u, 0}, u− = max{−u, 0}, and u are in W 1,p (Ω), and their generalized derivatives are given by (Di u+ )(x) = Di u(x) 0 if u(x) > 0 , if u(x) ≤ 0 , 0 −Di u(x) ⎧ ⎪ ⎨Di u(x) (Di u)(x) = 0 ⎪ ⎩ −Di u(x) (Di u− )(x) = if u(x) ≥ 0 , if u(x) < 0 , if u(x) > 0 , if u(x) = 0 , if u(x) < 0 . As for the traces of u+ and u− , we have (cf. [66]) γ(u+ ) = (γ(u))+ , γ(u− ) = (γ(u))− . Lemma 2.87 (Lattice Structure). Let u, v ∈ W 1,p (Ω), 1 ≤ p < ∞. Then max{u, v} and min{u, v} are in W 1,p (Ω) with generalized derivatives Di max{u, v}(x) = Di u(x) Di v(x) if u(x) > v(x) , if v(x) ≥ u(x) , Di min{u, v}(x) = Di u(x) Di v(x) if u(x) < v(x) , if v(x) ≤ u(x) . Proof: The assertion follows easily from the above examples and the generalized chain rule by using max{u, v} = (u−v)+ +v and min{u, v} = u−(u−v)+ ; see [112, Theorem 1.20]. Lemma 2.88. If (uj ), ( vj ) ⊂ W 1,p (Ω) (1 ≤ p < ∞) are such that uj → u and vj → v in W 1,p (Ω), then min{uj , vj } → min{u, v} and max{uj , vj } → max{u, v} in W 1,p (Ω) as j → ∞. For the proof, see [112, Lemma 1.22]. By means of Lemma 2.88, we readily obtain the following result. Lemma 2.89. Let u, ū ∈ W 1,p (Ω) satisfy u ≤ ū, and let T be the truncation operator deﬁned by ⎧ ⎪ ⎨ ū(x) if u(x) > ū(x) , T u(x) = u(x) if u(x) ≤ u(x) ≤ ū(x) , ⎪ ⎩ u(x) if u(x) < u(x) . Then T is a bounded continuous mapping from W 1,p (Ω) [respectively, Lp (Ω)] into itself. 36 2 Mathematical Preliminaries Proof: The truncation operator T can be represented in the form T u = max{u, u} + min{u, ū} − u. Thus, the assertion easily follows from Lemma 2.88. Lemma 2.90 (Lattice Structure). If u, v ∈ W01,p (Ω), then max{u, v} and min{u, v} are in W01,p (Ω). Lemma 2.90 implies that W01,p (Ω) has a lattice structure as well; see [112]. A partial ordering of traces on ∂Ω is given as follows. Deﬁnition 2.91. Let u ∈ W 1,p (Ω), 1 ≤ p < ∞. Then u ≤ 0 on ∂Ω if u+ ∈ W01,p (Ω). 2.2.4 Some Inequalities In this section, we recall some wellknown inequalities that are frequently used and that can be found in standard textbooks; see [93, 132, 222]. Young’s Inequality Let 1 < p, q < ∞, and 1/p + 1/q = 1. Then ap bq + p q ab ≤ (a, b ≥ 0). Proof: For a, b ∈ R+ satisfying ab = 0, the inequality is trivially satisﬁed. Let a, b > 0. As the function x → ex is convex, it follows that 1 p ab = elog a+log b = e p log a + q1 log bq ≤ bq 1 log ap 1 log bq ap e + + e = p q p q Young’s Inequality with Epsilon Let 1 < p, q < ∞, and 1/p + 1/q = 1. Then ab ≤ εap + C(ε)bq (a, b ≥ 0, ε > 0) with C(ε) = (εp)−q/p 1q . Proof: Again we only need to consider the case where a, b > 0. In this case, we set ab = ((εp)1/p a)( (εp)b1/p ) and apply Young’s inequality. 2.2 Sobolev Spaces 37 Equivalent Norms Let 1 ≤ s < ∞, and ξi ∈ R, ξi ≥ 0, i = 1, . . . , N, then we have the following inequality: N 1/s 1/s N N ξis a i=1 ≤ ξis ξi ≤ b i=1 , i=1 where a and b are some positive constants depending only on N and s. Proof: The inequality is an immediate consequence of the fact that all norms in RN are equivalent to each other. Monotonicity Inequality Let 1 < p < ∞. Consider the vectorvalued function a : RN → RN deﬁned by a(ξ) = ξp−2 ξ for ξ = 0, a(0) = 0. If 1 < p < 2, then we have (a(ξ) − a(ξ )) · (ξ − ξ ) > 0 for all ξ, ξ ∈ RN , ξ = ξ . If 2 ≤ p < ∞, then a constant c > 0 exists such that (a(ξ) − a(ξ )) · (ξ − ξ ) ≥ c ξ − ξ p for all ξ ∈ RN . Hölder’s Inequality Let 1 ≤ p, q ≤ ∞, 1 p + 1 q = 1. If u ∈ Lp (Ω), v ∈ Lq (Ω), then one has Ω uv dx ≤ uLp (Ω) vLq (Ω) . Minkowski’s Inequality Let 1 ≤ p ≤ ∞ and u, v ∈ Lp (Ω); then u + vLp (Ω) ≤ uLp (Ω) + vLp (Ω) . 38 2 Mathematical Preliminaries Clarkson’s Inequalities Let u, v ∈ Lp (Ω). If 2 ≤ p < ∞, then u + vpLp (Ω) + u − vpLp (Ω) ≤ 2p−1 upLp (Ω) + vpLp (Ω) . If 1 < p < 2, then u + vpLp (Ω) + u − vpLp (Ω) ≤ 2 upLp (Ω) + vpLp (Ω) . Proof: Use the function ϕ : [0, 1] → R deﬁned by ϕ(t) = (1 + t)p + (1 − t)p , 1 + tp t ∈ [0, 1]. Remark 2.92. It follows immediately from Clarkson’s inequalities that the spaces Lp (Ω) and the Sobolev spaces W m,p (Ω) are uniformly convex for 1 < p < ∞, and m = 0, 1, . . . , . Poincaré–Friedrichs Inequality Let Ω ⊂ RN be a bounded domain, 1 ≤ p < ∞, and u ∈ W01,p (Ω). Then we have the estimate uLp (Ω) ≤ C ∇uLp (Ω) , where the constant C only depends on p, N, and Ω. Remark 2.93. The Poincaré–Friedrichs inequality implies that uW 1,p (Ω) = ∇uLp (Ω) 0 deﬁnes an equivalent norm on W01,p (Ω). Equivalent norms on W 1,p (Ω) play an important role in the treatment of boundary value problems. The following general result provides a tool to identify equivalent norms on W 1,p (Ω). Proposition 2.94. Let Ω ⊂ RN , N ≥ 1, be a bounded domain with Lipschitz boundary ∂Ω. Assume ϕ : W 1,p (Ω) → R+ , 1 ≤ p < ∞, is a seminorm that satisﬁes the following conditions: (i) A positive constant d exists such that ϕ(u) ≤ d uW 1,p (Ω) for all u ∈ W 1,p (Ω). (ii) If u = constant, then ϕ(u) = 0 implies u = 0. 2.3 Operators of Monotone Type 39 Then · ∼ deﬁned by p1 u∼ = ∇upLp (Ω) + ϕ(u)p deﬁnes an equivalent norm in W 1,p (Ω). As an application of Proposition 2.94, we obtain, e.g., an equivalent norm on the closed subspace VΓ of W 1,p (Ω) deﬁned by VΓ = {u ∈ W 1,p (Ω) : γ(u) = 0 on Γ }, where Γ ⊂ ∂Ω is some part of the boundary ∂Ω with strictly positive surface measure Γ  > 0. To this end, deﬁne ϕ by ϕ(u) = Γ p γ(u) dΓ p1 for all u ∈ W 1,p (Ω), where γ is the trace operator. We observe that (i) and (ii) of Proposition 2.94 are satisﬁed, and thus · ∼ deﬁned above gives an equivalent norm on W 1,p (Ω). As ϕ(u) = 0 for u ∈ VΓ , we see that u∼ = ∇uLp (Ω) for all u ∈ VΓ is an equivalent norm on the subspace VΓ . 2.3 Operators of Monotone Type In this section, we provide the basic results on pseudomonotone operators from a Banach space X into its dual space X ∗ . 2.3.1 Main Theorem on Pseudomonotone Operators Let X be a real, reﬂexive Banach space with norm · , X ∗ its dual space, and denote by ·, · the duality pairing between them. The norm convergence in X and X ∗ is denoted by “→” and the weak convergence by “ ”. Deﬁnition 2.95. Let A : X → X ∗ ; then A is called (i) continuous (respectively, weakly continuous) iﬀ un → u implies Aun → u implies Aun Au) Au (respectively, un (ii) demicontinuous iﬀ un → u implies Aun Au (iii) hemicontinuous iﬀ the real function t → A(u + tv), w is continuous on [0, 1] for all u, v, w ∈ X (iv) strongly continuous or completely continuous iﬀ un u implies Aun → Au 40 2 Mathematical Preliminaries (v) bounded iﬀ A maps bounded sets into bounded sets (vi) coercive iﬀ limu→∞ Au,u u = +∞ Deﬁnition 2.96 (Operators of Monotone Type). Let A : X → X ∗ ; then A is called (i) monotone (respectively, strictly monotone) iﬀ Au − Av, u − v ≥ (respectively, >) 0 for all u, v ∈ X with u = v (ii) strongly monotone iﬀ there is a constant c > 0 such that Au−Av, u−v ≥ cu − v2 for all u, v ∈ X (iii) uniformly monotone iﬀ Au − Av, u − v ≥ a(u − v)u − v for all u, v ∈ X where a : [0, ∞) → [0, ∞) is strictly increasing with a(0) = 0 and a(s) → +∞ as s → ∞ (iv) pseudomonotone iﬀ un u and lim supn→∞ Aun , un − u ≤ 0 implies Au, u − w ≤ lim inf n→∞ Aun , un − w for all w ∈ X u and lim supn→∞ Aun , un − u ≤ 0 (v) to satisfy (S+ )condition iﬀ un imply un → u We can show (cf. [18]) that the pseudomonotonicity according to (iv) of Definition 2.96 is equivalent to the following deﬁnition. Deﬁnition 2.97. The operator A : X → X ∗ is pseudomonotone iﬀ un u and lim supn→∞ Aun , un − u ≤ 0 implies Aun Au and Aun , un → Au, u . For the following result, see [222, Proposition 27.6]. Lemma 2.98. Let A, B : X → X ∗ be operators on the real reﬂexive Banach space X. Then the following implications hold: (i) If A is monotone and hemicontinuous, then A is pseudomonotone. (ii) If A is strongly continuous, then A is pseudomonotone. (iii) If A and B are pseudomonotone, then A + B is pseudomonotone. The main theorem on pseudomonotone operators due to Brézis is given by the next theorem (see [222, Theorem 27.A]). Theorem 2.99 (Main Theorem on Pseudomonotone Operators). Let X be a real, reﬂexive Banach space, and let A : X → X ∗ be a pseudomonotone, bounded, and coercive operator, and b ∈ X ∗ . Then a solution of the equation Au = b exists. Remark 2.100. Theorem 2.99 contains several important surjectivity results as special cases, such as Lax–Milgram’s theorem and the Main Theorem on Monotone Operators, which will be formulated in the following corollaries. 2.3 Operators of Monotone Type 41 Corollary 2.101 (Main Theorem on Monotone Operators). Let X be a real, reﬂexive Banach space, and let A : X → X ∗ be a monotone, hemicontinuous, bounded, and coercive operator, and b ∈ X ∗ . Then a solution of the equation Au = b exists. For the proof of Corollary 2.101, we have only to mention that in view of Lemma 2.98, a monotone and hemicontinuous operator is pseudomonotone. Corollary 2.102 (Lax–Milgram’s Theorem). Let X be a real Hilbert space, and let a : X × X → R be a bilinear form. Assume that (i) a is bounded; i.e., there is a C > 0 such that a(x, y) ≤ Cxy for x, y ∈ X. (ii) a is coercive, i.e., there is a C0 > 0 such that a(x, x) ≥ C0 x2 for x ∈ H. Then, for each f in X ∗ , there is a unique element u in X such that a(u, v) = f, v for v ∈ X. The mapping f → u is onetoone, continuous, and linear from X ∗ onto X. As for the proof, note that the bilinear form a of Corollary 2.102 deﬁnes a linear, bounded, and strongly monotone operator A : X → X ∗ acccording to Au, v = a(u, v) for all u, v ∈ X, and thus the equation a(u, v) = f, v of Corollary 2.102 is equivalent with the operator equation Au = f in X ∗ . The existence result for the latter follows immediately from Corollary 2.101, because A is strongly monotone and continuous and therefore, in particular, also coercive. The uniqueness is a consequence of the strong monotonicity of A. 2.3.2 Leray–Lions Operators An important class of operators of monotone type is the socalled Leray–Lions operators (see [215, 152]). These kinds of operators occur in the functional analytical treatment of nonlinear elliptic and parabolic problems. Deﬁnition 2.103 (Leray–Lions Operator). Let X be a real, reﬂexive Banach space. We say that A : X → X ∗ is a Leray–Lions operator if it is bounded and satisﬁes Au = A(u, u), for u ∈ X, where A : X × X → X ∗ has the following properties: 42 2 Mathematical Preliminaries (i) For any u ∈ X, the mapping v → A(u, v) is bounded and hemicontinuous from X to its dual X ∗ , with A(u, u) − A(u, v), u − v ≥ 0 for v ∈ X. (ii) For any v ∈ X, the mapping u → A(u, v) is bounded and hemicontinuous from X to its dual X ∗ . (iii) For any v ∈ X, A(un , v) converges weakly to A(u, v) in X ∗ if (un ) ⊂ X is such that un u in X and A(un , un ) − A(un , u), un − u → 0. (iv) For any v ∈ X, A(un , v), un converges to F, u if (un ) ⊂ V is such that un u in X, and A(un , v) F in X ∗ . As for the proof of the next theorem, see [215]. Theorem 2.104. Every Leray–Lions operator A : X → X ∗ is pseudomonotone. Next we will see that quasilinear elliptic operators satisfying certain structure and growth conditions represent Leray–Lions operators. To this end, we need to study ﬁrst the mapping properties of superposition operators, which are also called Nemytskij operators. Deﬁnition 2.105 (Nemytskij Operator). Let Ω ⊂ RN , N ≥ 1, be a nonempty measurable set, and let f : Ω × Rm → R, m ≥ 1, and u : Ω → Rm be a given function. Then the superposition or Nemytskij operator F assigns u → f ◦ u; i.e., F is given by F u(x) = (f ◦ u)(x) = f (x, u(x)) for x ∈ Ω. Deﬁnition 2.106 (Carathéodory Function). Let Ω ⊂ RN , N ≥ 1, be a nonempty measurable set, and let f : Ω × Rm → R, m ≥ 1. The function f is called a Carathéodory function if the following two conditions are satisﬁed: (i) x → f (x, s) is measurable in Ω for all s ∈ Rm . (ii) s → f (x, s) is continuous on Rm for a.e. x ∈ Ω. Lemma 2.107. Let f : Ω × Rm → R, m ≥ 1, be a Carathéodory function that satisﬁes a growth condition of the form m f (x, s) ≤ k(x) + c si pi /q , ∀ s = (s1 , . . . , sm ) ∈ Rm , a.e. x ∈ Ω, i=1 for some positive constant c and some k ∈ Lq (Ω), and 1 ≤ q, pi < ∞ for all i = 1, . . . , m. Then the Nemytskij operator F deﬁned by 2.3 Operators of Monotone Type 43 F u(x) = f (x, u1 (x), . . . , um (x)) is continuous and bounded from Lp1 (Ω) × · · · × Lpm (Ω) into Lq (Ω). Here u denotes the vector function u = (u1 , . . . , um ). Furthermore, m F uLq (Ω) ≤ c p /q kLq (Ω) + i=1 ui Lipi (Ω) . Deﬁnition 2.108. Let Ω ⊂ RN , N ≥ 1, be a nonempty measurable set. A function f : Ω × Rm → R, m ≥ 1, is called superpositionally measurable (or supmeasurable) if the function x → F u(x) is measurable in Ω whenever the component functions ui : Ω → R of u = (u1 , . . . , um ) are measurable. Now let Ω ⊂ RN be a bounded domain with Lipschitz boundary ∂Ω, let A1 be the secondorder quasilinear diﬀerential operator in divergence form given by N ∂ A1 u(x) = − ai (x, u(x), ∇u(x)), ∂xi i=1 and let A0 denote the operator A0 u(x) = a0 (x, u(x), ∇u(x)) . Let 1 < p < ∞, 1/p + 1/q = 1, and assume for the coeﬃcients ai : Ω × R × RN → R, i = 0, 1, . . . , N the following conditions. (H1) Carathéodory and Growth Condition: Each ai (x, s, ξ) satisﬁes Carathéodory conditions, i.e., is measurable in x ∈ Ω for all (s, ξ) ∈ R × RN and continuous in (s, ξ) for a.e. x ∈ Ω. A constant c0 > 0 and a function k0 ∈ Lq (Ω) exist so that ai (x, s, ξ) ≤ k0 (x) + c0 (sp−1 + ξp−1 ) for a.e. x ∈ Ω and for all (s, ξ) ∈ R×RN , with ξ denoting the Euclidian norm of the vector ξ. (H2) Monotonicity Type Condition: The coeﬃcients ai satisfy a monotonicity condition with respect to ξ in the form N (ai (x, s, ξ) − ai (x, s, ξ ))(ξi − ξi ) > 0 i=1 for a.e. x ∈ Ω , for all s ∈ R, and for all ξ, ξ ∈ RN with ξ = ξ . (H3) Coercivity Type Condition: N ai (x, s, ξ)ξi ≥ νξp − k(x) i=1 for a.e. x ∈ Ω , for all s ∈ R, and for all ξ ∈ RN with some constant ν > 0 and some function k ∈ L1 (Ω). 44 2 Mathematical Preliminaries Let V be a closed subspace of W 1,p (Ω) such that W01,p (Ω) ⊂ V ⊂ W 1,p (Ω), then under condition (H1) the diﬀerential operators A1 and A0 generate mappings from V into its dual space (again denoted by A1 and A0 , respectively) deﬁned by N A1 u, ϕ = i=1 ∂ϕ ai (x, u, ∇u) dx , ∂x i Ω A0 u, ϕ = Ω a0 (x, u, ∇u) ϕ dx . Theorem 2.109. Set A = A1 + A0 . Then the operators A, A0 , and A1 have the following properties: (i) If (H1) is satisﬁed, then the mappings A, A1 , A0 : V → V ∗ are continuous and bounded. (ii) If (H1) and (H2) are satisﬁed, then A : V → V ∗ is pseudomonotone. (iii) If (H1), (H2), and (H3) are satisﬁed, then A has the (S+ )property. Conditions (H1) and (H2) are the socalled Leray–Lions conditions that guarantee that A is pseudomonotone. In their original paper, Leray and Lions [149] showed the pseudomonotonicity under conditions (H1), (H2), and the following additional condition. N i (x,s,ξ)ξi (H4) lim supξ→∞, s∈B i=1 aξ+ξ p−1 = +∞, for a.e. x ∈ Ω and all bounded sets B. However, Landes and Mustonen have shown in [136] that condition (H4) is redundant for the pseudomonotonicity of A. As for the proof of the results stated in Theorem 2.109 as well as on existence theorems involving pseudomonotone operators, we refer to [17, 18] and [23, 27, 105, 152, 208, 222]. Example 2.110. Let Ω ⊂ RN be a bounded domain. A prototype of a monotone elliptic operator in Ω is the negative of the pLaplacian Δp , 1 < p < ∞, deﬁned by Δp u = div(∇up−2 ∇u) where ∇u = (∂u/∂x1 , . . . , ∂u/∂xN ). This operator coincides with the Laplacian Δ if p = 2, and is of the form A1 with the coeﬃents ai , i = 1, . . . , N, given by ai (x, s, ξ) = ξp−2 ξi . Thus, hypothesis (H1) is satisﬁed with k0 = 0, c0 = 1, and a0 = 0. Hypothesis (H2) follows from the inequalities satisﬁed by the vectorvalued function ξ → ξp−2 ξ, (see Sect. 2.2.4) and (H3) is obviously true with ν = 1 and k = 0 due to N N ai (x, s, ξ)ξi = i=1 ξp−2 ξi ξi = ξp . i=1 Therefore, hypotheses (H1)–(H3) are satisﬁed by the negative pLaplacian, and in view of Theorem 2.109, we see that −Δp : V → V ∗ is continuous, 2.3 Operators of Monotone Type 45 bounded, pseudomonotone, and has the (S+ )property. Moreover, from the inequality −Δp u − (−Δp v), u − v = (∇up−2 ∇u − ∇vp−2 ∇v)(∇u − ∇v) dx ≥ 0, Ω for all u, v ∈ V, we infer that −Δp : V → V ∗ is, in particular, also a monotone operator. Depending on the domain of deﬁnition of −Δp , we can say even more. For example, let V = W01,p (Ω). According to Sect. 2.2.4, uV = Ω p 1/p ∇u dx deﬁnes an equivalent norm in V . From the inequalities for the function ξ → ξp−2 ξ, we see that the operator −Δp : W01,p (Ω) → (W01,p (Ω))∗ has the mapping properties given in the following lemma. Lemma 2.111. Let V be a closed subspace of W 1,p (Ω) such that W01,p (Ω) ⊂ V ⊂ W 1,p (Ω). Then one has: (i) −Δp : V → V ∗ is continuous, bounded, pseudomonotone, and has the (S+ )property. (ii) −Δp : W01,p (Ω) → (W01,p (Ω))∗ is (a) strictly monotone if 1 < p < ∞. (b) strongly monotone if p = 2 (Laplacian). (c) uniformly monotone if 2 < p < ∞. 2.3.3 Multivalued Pseudomonotone Operators In this section, we brieﬂy recall the main results of the theory of pseudomonotone multivalued operators developed by Browder and Hess to the extent it will be needed in the study of variational and hemivariational inequalities. For the proofs and a more detailed presentation, we refer to the monographs [222, 177]. First we present basic results about the continuity of multivalued functions (multifunctions) and provide useful equivalent descriptions of these notions. Even though these notions can be deﬁned in a much more general context, we conﬁne ourselves to mappings between Banach spaces, which is suﬃcient for our purpose. Deﬁnition 2.112 (Semicontinuous Multifunctions). Let X, Y be Banach spaces and A : X → 2Y be a multifunction. (i) A is called upper semicontinuous at x0 , if for every open subset V ⊂ Y with A(x0 ) ⊂ V, a neighborhood U (x0 ) exists such that A(U (x0 )) ⊂ V. If A is upper semicontinuous at every x0 ∈ X, we call A upper semicontinuous in X. 46 2 Mathematical Preliminaries (ii) A is called lower semicontinuous at x0 if for every neighborhood V (y) of every y ∈ A(x0 ), a neighborhood U (x0 ) exists such that A(u) ∩ V (y) = ∅ for all u ∈ U (x0 ). If A is lower semicontinuous at every x0 ∈ X, we call A lower semicontinuous in X. (iii) A is called continuous at x0 if A is both upper and lower semicontinuous at x0 . If A is continuous at every x0 ∈ X, we call A continuous in X. Alternative equivalent continuity criteria are given in the following propositions. To this end, we introduce the preimage of a multifunction. Deﬁnition 2.113 (Preimage). Let M ⊂ Y and A : X → 2Y be a multifunction. The preimage A−1 (M ) is deﬁned by A−1 (M ) = {x ∈ X : A(x) ∩ M = ∅}. Proposition 2.114. Let X, Y be Banach spaces and A : X → 2Y be a multifunction. Then the following statements are equivalent: (i) A is upper semicontinuous. (ii) For all closed sets C ⊂ Y, the preimage A−1 (C) is closed. (iii) If x ∈ X, (xn ) is a sequence in X with xn → x as n → ∞, and V is an open set in Y such that A(x) ⊂ V , then n0 ∈ N exists depending on V such that for all n ≥ n0 , we have A(xn ) ⊂ V. Proposition 2.115. Let X, Y be Banach spaces and A : X → 2Y be a multifunction. Then the following statements are equivalent: (i) A is lower semicontinuous. (ii) For all open sets O ⊂ Y, the preimage A−1 (O) is open. (iii) If x ∈ X, (xn ) is a sequence in X with xn → x as n → ∞, and y ∈ A(x), then for every n ∈ N, we can ﬁnd a yn ∈ A(xn ), such that yn → y, as n → ∞. Remark 2.116. For a singlevalued operator A : X → Y , upper semicontinuous and lower semicontinuous in the multivalued setting is identical with continuous. For A : M → 2N having the same corresponding properties, where M and N are subsets of the Banach spaces X and Y, respectively, then M and N have to be equipped with the induced topology. Next we introduce the notion of multivalued monotone and pseudomonotone operators from a real, reﬂexive Banach space X into its dual space and formulate the main surjectivity result for these kinds of operators. Deﬁnition 2.117 (Graph). Let X be a real Banach space, and let A : X → ∗ 2X be a multivalued mapping; i.e., to each u ∈ X, there is assigned a subset 2.3 Operators of Monotone Type 47 A(u) of X ∗ , which may be empty if u ∈ / D(A), where D(A) is the domain of A given by D(A) = {u ∈ X : A(u) = ∅}. The graph of A denoted by Gr(A) is given by Gr(A) = {(u, u∗ ) ∈ X × X ∗ : u∗ ∈ A(u)}. ∗ Deﬁnition 2.118 (Monotone Operator). The mapping A : X → 2X is called (i) monotone iﬀ u∗ − v ∗ , u − v ≥ 0 for all (u, u∗ ), (v, v ∗ ) ∈ Gr(A) (ii) strictly monotone iﬀ u∗ − v ∗ , u − v > 0 for all (u, u∗ ), (v, v ∗ ) ∈ Gr(A), u = v (iii) maximal monotone iﬀ A is monotone and there is no monotone mapping ∗ Ã : X → 2X such that Gr(A) is a proper subset of Gr(Ã), which is equivalent to the following implication: (u, u∗ ) ∈ X × X ∗ : u∗ − v ∗ , u − v ≥ 0 for all (v, v ∗ ) ∈ Gr(A) implies (u, u∗ ) ∈ Gr(A) The notions of strongly and uniformly monotone multivalued operators are deﬁned in a similar way as for singlevalued operators. Example 2.119. If X = R, then a maximal monotone mapping β : R → 2R is called maximal monotone graph in R2 . For example, an increasing function f : R → R generates a maximal monotone graph β in R2 given by β(s) := [f (s − 0), f (s + 0)], where f (s ± 0) are the onesided limits of f in s. A singlevalued operator A : D(A) ⊂ X → X ∗ is to be understood as a multivalued operator A : X → X ∗ by setting Au = {Au} if u ∈ D(A) and Au = ∅ otherwise. Thus, A is monotone iﬀ Au − Av, u − v ≥ 0 for all u, v ∈ D(A), and A : D(A) ⊂ X → X ∗ is maximal monotone iﬀ A is monotone and the condition (u, u∗ ) ∈ X × X ∗ : u∗ − Av, u − v ≥ 0 for all v ∈ D(A) implies u ∈ D(A) and u∗ = Au. 48 2 Mathematical Preliminaries Deﬁnition 2.120 (Pseudomonotone Operator). Let X be a real reﬂexive ∗ Banach space. The operator A : X → 2X is called pseudomonotone if the following conditions hold: (i) The set A(u) is nonempty, bounded, closed, and convex for all u ∈ X. (ii) A is upper semicontinuous from each ﬁnitedimensional subspace of X to the weak topology on X ∗ . (iii) If (un ) ⊂ X with un u, and if u∗n ∈ A(un ) is such that lim sup u∗n , un − u ≤ 0, then to each element v ∈ X, u∗ (v) ∈ A(u) exists with lim inf u∗n , un − v ≥ u∗ (v), u − v . Deﬁnition 2.121 (Generalized Pseudomonotone Operator). Let X be ∗ a real reﬂexive Banach space. The operator A : X → 2X is called generalized pseudomonotone if the following holds: u in X and u∗n u∗ Let (un ) ⊂ X and (u∗n ) ⊂ X ∗ with u∗n ∈ A(un ). If un ∗ ∗ ∗ in X and if lim sup un , un − u ≤ 0, then the element u lies in A(u) and u∗n , un → u∗ , u . The next two propositions provide the relation between pseudomonotone and generalized pseudomontone operators. Proposition 2.122. Let X be a real reﬂexive Banach space. If the operator ∗ A : X → 2X is pseudomonotone, then A is generalized pseudomonotone. Under the additional assumption of boundedness, the following converse of Proposition 2.122 is true. Proposition 2.123. Let X be a real reﬂexive Banach space, and assume that ∗ A : X → 2X satisﬁes the following conditions: (i) For each u ∈ X, we have that A(u) is a nonempty, closed, and convex subset of X ∗ . ∗ (ii) A : X → 2X is bounded. (iii) If un u in X and u∗n u∗ in X ∗ with u∗n ∈ A(un ) and if ∗ lim sup un , un − u ≤ 0, then u∗ ∈ A(u) and u∗n , un → u∗ , u . ∗ Then the operator A : X → 2X is pseudomonotone. As for the proof of Proposition 2.123 we refer to [177, Chap. 2]. Note that the notion of boundedness of a multivalued operator is exactly the same as for singlevalued operators; i.e., the image of a bounded set is again bounded. The relation between maximal monotone and pseudomonotone operators as well as the invariance of pseudomonotonicity under addition is given in the following theorem. 2.4 FirstOrder Evolution Equations 49 Theorem 2.124. Let X be a real reﬂexive Banach space, and let A, Ai : X → ∗ 2X , i = 1, 2. (i) If A is maximal monotone with D(A) = X, then A is pseudomonotone. (ii) If A1 and A2 are two pseudomonotone operators, then the sum A1 + A2 : ∗ X → 2X is pseudomonotone. The main theorem on pseudomonotone multivalued operators is formulated in the next theorem. Theorem 2.125. Let X be a real reﬂexive Banach space, and let A : X → ∗ 2X be a pseudomonotone and a bounded operator, which is coercive in the sense that a realvalued function c : R+ → R exists with c(r) → +∞, as r → +∞ such that for all (u, u∗ ) ∈ Gr(A), we have u∗ , u − u0 ≥ c(uX )uX for some u0 ∈ X. Then A is surjective; i.e., range(A) = X. Remark 2.126. We remark that the boundedness condition supposed in Theorem 2.125 can be dropped (see [177, Theorem 2.6]). This is because by deﬁniti