1

Direct Methods in the Calculus of Variations Springer Berlin Heidelberg Bernard Dacorogna (auth.) İl: 1989 Dil: english Fayl: PDF, 3.61 MB Sizin teqləriniz: 0 / 0

2

Unbounded functionals in the calculus of variations Chapman and Hall/CRC Luciano Carbone, Riccardo De Arcangelis İl: 2001 Dil: english Fayl: DJVU, 2.33 MB Sizin teqləriniz: 0 / 0

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
Martin-Luther-Universität
Halle-Wittenberg
D-06099 Halle
Germany
siegfried.carl@mathematik.uni-halle.de

Vy Khoi Le
Department of Mathematics and
Statistics
University of Missouri-Rolla
Rolla, MO 65409
U.S.A
vy@umr.edu

Dumitru Motreanu
Département de Mathématiques
Université de Perpignan
66860 Perpignan
France
motreanu@univ-perp.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
ISBN-13: 978-0-387-30653-7

e-ISBN-13: 978-0-387-46252-3

Printed on acid-free 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
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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 sub-supersolution 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 self-contained. 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 First-Order Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Vector-Valued 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 p-Laplacian . . . . . . . . . . . . . . . . . 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 Sign-Changing 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 Sub-Supersolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Sub-Supersolutions . . . . . . . . . . . . . . 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 Sub-Supersolutions 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 Sub-Supersolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Sub-Supersolutions 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 sub-supersolutions 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 order-preserving
property of solutions of (1.1), which means that if u1 and u2 are the solutions

2

1 Introduction

of (1.1) corresponding to right-hand 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 (|∇u|p−2 ∇u) is the p-Laplacian, 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 :
|∇u|p−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 dual-order 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 well-known equivalent definition of a weak
supersolution of (1.7). The function ū ∈ V is a weak supersolution if ū ≥ 0
on ∂Ω and


|∇ū|p−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. Sub-supersolution

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 sub-supersolutions 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 sub-supersolutions 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 Borel-measurable, 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 sub-supersolution 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 sub-supersolutions 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
left-sided and right-sided 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) =
|∇u|p 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 well-known 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. Zig-zag 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 variational-hemivariational 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 (so-called 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 variational-hemivariational 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 well-known 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
sub-supersolution 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 self-contained. 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 sub-supersolution 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 sub-supersolution
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 self-contained 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 fixed-point theorem from Schauder holds.
Theorem 2.2 (Schauder’s Fixed-Point 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 finite-dimensional normed linear spaces, Theorem 2.2 reduces to Brouwer’s
fixed-point theorem.
Corollary 2.3 (Brouwer’s Fixed-Point 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 finite-dimensional 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 graph-closed)
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 so-called 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 graph-closed. 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 finite-dimensional Banach spaces, closed and bounded sets are compact.
This result is no longer true for infinite-dimensional 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 finite-dimensional.
According to Theorem 2.26, in infinite-dimensional Banach spaces, there
are bounded sequences that have no convergent subsequence. This lack of compactness in infinite-dimensional 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 finite-dimensional, 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 so-called 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 well-known 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 Gâteaux-differentiable
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 real-valued functions in Rn to real-valued
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 Gâteaux-differentiable 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 Fréchet-differentiable 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 Fréchet-differentiable at u ∈ U, then f is Gâteaux-differentiable at
u.
(ii) If f is Gâteaux-differentiable in a neighborhood of u0 and DG f is continuous at u0 , then f is Fréchet-differentiable at u0 and f  (u0 ) = DG f (u0 ).
Remark 2.46. If f : U ⊂ X → Y is Fréchet-differentiable 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

|u|p−2 uh dx

if we consider real-valued functions u : Ω → R. In case the functions are
complex-valued, we get

p
δf (u; h) =
|u|p−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 right-hand 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 Gâteaux-differentiable 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 Gâteaux-differentiable 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 one-sided
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 Lebesgue-measurable 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) well-ordered 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

|u|p 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 multi-index,
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).

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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 Lipschitz-manifold) 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.

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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 Borel-measurable 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 well-known 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 vector-valued function a : RN → RN defined by
a(ξ) = |ξ|p−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) .

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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 one-to-one, 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 so-called 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
sup-measurable) 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 second-order 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 (|s|p−1 + |ξ|p−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 ≥ ν|ξ|p − 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).

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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 so-called 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 sup|ξ|→∞, s∈B i=1 a|ξ|+|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 p-Laplacian Δp , 1 < p < ∞,
defined by
Δp u = div(|∇u|p−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) = |ξ|p−2 ξi .
Thus, hypothesis (H1) is satisfied with k0 = 0, c0 = 1, and a0 = 0. Hypothesis
(H2) follows from the inequalities satisfied by the vector-valued function ξ →
|ξ|p−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

|ξ|p−2 ξi ξi = |ξ|p .

i=1

Therefore, hypotheses (H1)–(H3) are satisfied by the negative p-Laplacian,
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 =
(|∇u|p−2 ∇u − |∇v|p−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 →
|ξ|p−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 single-valued 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
∗
à : 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 single-valued 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 one-sided limits of f in s.
A single-valued 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 finite-dimensional 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
single-valued 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 First-Order 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 real-valued 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