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Nonsmooth Variational Problems and Their Inequalities: Comparison Principles and Applications 2007 ed. [Hardback]

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  • Format: Hardback, 398 pages, height x width: 235x155 mm, weight: 1650 g, X, 398 p., 1 Hardback
  • Series: Springer Monographs in Mathematics
  • Pub. Date: 09-Nov-2006
  • Publisher: Springer-Verlag New York Inc.
  • ISBN-10: 0387306536
  • ISBN-13: 9780387306537
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Nonsmooth Variational Problems and Their Inequalities: Comparison Principles and Applications 2007 ed.Larger Image
  • Format: Hardback, 398 pages, height x width: 235x155 mm, weight: 1650 g, X, 398 p., 1 Hardback
  • Series: Springer Monographs in Mathematics
  • Pub. Date: 09-Nov-2006
  • Publisher: Springer-Verlag New York Inc.
  • ISBN-10: 0387306536
  • ISBN-13: 9780387306537
Other books in subject:
Permanent link: https://www.kriso.ee/db/9780387306537.html
Keywords:
This monograph focuses primarily on nonsmooth variational problems that arise from boundary value problems with nonsmooth data and/or nonsmooth constraints, such as is multivalued elliptic problems, variational inequalities, hemivariational inequalities, and their corresponding evolution problems. The main purpose of this book is to provide a systematic and unified exposition of comparison principles based on a suitably extended sub-supersolution method. This method is an effective and flexible technique to obtain existence and comparison results of solutions. Also, it can be employed for the investigation of various qualitative properties, such as location, multiplicity and extremality of solutions. In the treatment of the problems under consideration a wide range of methods and techniques from nonlinear and nonsmooth analysis is applied, a brief outline of which has been provided in a preliminary chapter in order to make the book self-contained.This text is an invaluable reference for researchers and graduate students in mathematics (functional analysis, partial differential equations, elasticity, applications in materials science and mechanics) as well as physicists and engineers.

This monograph focuses primarily on nonsmooth variational problems that arise from boundary value problems with nonsmooth data and/or nonsmooth constraints, such as multivalued elliptic problems, variational inequalities, hemivariational inequalities, and their corresponding evolution problems. It provides a systematic and unified exposition of comparison principles based on a suitably extended sub-supersolution method.

Reviews

From the reviews:









"This monograph presents in a systematic way the method of sub- and supersolutions for solving variational and hemivariational inequalities. Each chapter begins with a short overview presenting cases and ideas, and concludes with notes and remarks giving references to the literature. It is carefully written and suitable for advanced graduate students and researchers having a good background in functional analysis, basic partial differential equations, and critical point theory." (Thomas Bartsch, Mathematical Reviews, Issue, 2007 i)



"This monograph contains seven chapters, the bibliography of 223 entries, and an index. This well-written book contains large number of material. It can be useful for graduate students and researchers interested in variational methods. In the beginning of each chapter the authors give some motivation for the material of the chapter." (Alexander G. Ramm, Zentralblatt MATH, Vol. 1109 (11), 2007)



"Thepresent monograph develops in a careful and thorough way the theory of sub- and supersolutions for variational equalities and inequalities. The book is an important addition to the literature on nonlinear analysis, convex analysis, variational and hemivariational inequalities, nonlinear elliptic and parabolic partial differential equations, elasticity theory, fracture mechanics, and general obstacle and unilateral problems; it will be a welcome addition to the libraries of researchers and students of these areas." (Klaus Schmitt, SIAM Review, Vol. 49 (3), 2007)

Preface v
1 Introduction 1(10)
2 Mathematical Preliminaries 11(70)
2.1 Basic Functional Analysis
11(17)
2.1.1 Operators in Normed Linear Spaces
11(4)
2.1.2 Duality in Banach Spaces
15(5)
2.1.3 Convex Analysis and Calculus in Banach Spaces
20(7)
2.1.4 Partially Ordered Sets
27(1)
2.2 Sobolev Spaces
28(11)
2.2.1 Spaces of Lebesgue Integrable Functions
28(2)
2.2.2 Definition of Sobolev Spaces
30(4)
2.2.3 Chain Rule and Lattice Structure
34(2)
2.2.4 Some Inequalities
36(3)
2.3 Operators of Monotone Type
39(10)
2.3.1 Main Theorem on Pseudomonotone Operators
39(2)
2.3.2 Leray—Lions Operators
41(4)
2.3.3 Multivalued Pseudomonotone Operators
45(4)
2.4 First-Order Evolution Equations
49(14)
2.4.1 Motivation
50(3)
2.4.2 Vector-Valued Functions
53(2)
2.4.3 Evolution Triple and Generalized Derivative
55(4)
2.4.4 Existence Results for Evolution Equations
59(3)
2.4.5 Multivalued Evolution Equations
62(1)
2.5 Nonsmooth Analysis
63(18)
2.5.1 Clarke's Generalized Gradient
63(5)
2.5.2 Some Calculus
68(5)
2.5.3 Critical Point Theory
73(4)
2.5.4 Linking Theorem
77(4)
3 Variational Equations 81(62)
3.1 Semilinear Elliptic Equations
81(12)
3.1.1 Comparison Principle
82(2)
3.1.2 Directed and Compact Solution Set
84(7)
3.1.3 Extremal Solutions
91(2)
3.2 Quasilinear Elliptic Equations
93(12)
3.2.1 Comparison Principle
94(3)
3.2.2 Directed and Compact Solution Set
97(6)
3.2.3 Extremal Solutions
103(2)
3.3 Quasilinear Parabolic Equations
105(18)
3.3.1 Parabolic Equation with p-Laplacian
110(2)
3.3.2 Comparison Principle for Quasilinear Equations
112(4)
3.3.3 Directed and Compact Solution Set
116(6)
3.3.4 Extremal Solutions
122(1)
3.4 Sign-Changing Solutions via Fueik Spectrum
123(11)
3.4.1 Introduction
124(1)
3.4.2 Preliminaries
125(5)
3.4.3 Main Result
130(4)
3.5 Quasilinear Elliptic Problems of Periodic Type
134(7)
3.5.1 Problem Setting
134(2)
3.5.2 Sub-Supersolutions
136(2)
3.5.3 Existence Result
138(3)
3.6 Notes and Comments
141(2)
4 Multivalued Variational Equations 143(68)
4.1 Motivation and Introductory Examples
143(12)
4.1.1 Motivation
144(2)
4.1.2 Comparison Principle: Subdifferential Case
146(3)
4.1.3 Comparison Principle: Clarke's Gradient Case
149(6)
4.2 Inclusions with Global Growth on Clarke's Gradient
155(12)
4.2.1 Preliminaries
157(3)
4.2.2 Comparison and Compactness Results
160(7)
4.3 Inclusions with Local Growth on Clarke's Gradient
167(13)
4.3.1 Comparison Principle
167(9)
4.3.2 Compactness and Extremality Results
176(4)
4.4 Application: Difference of Multifunctions
180(10)
4.4.1 Hypotheses and Main Result
181(1)
4.4.2 A Priori Bounds
182(4)
4.4.3 Proof of Theorem 4.36
186(4)
4.5 Parabolic Inclusions with Local Growth
190(18)
4.5.1 Comparison Principle
191(10)
4.5.2 Extremality and Compactness Results
201(7)
4.6 An Alternative Concept of Sub-Supersolutions
208(1)
4.7 Notes and Comments
209(2)
5 Variational Inequalities 211(68)
5.1 Variational Inequalities on Closed Convex Sets
213(21)
5.1.1 Solutions and Extremal Solutions above Subsolutions
214(12)
5.1.2 Comparison Principle and Extremal Solutions
226(8)
5.2 Variational Inequalities with Convex Functionals
234(12)
5.2.1 General Settings- Sub- and Supersolutions
235(3)
5.2.2 Existence and Comparison Results
238(4)
5.2.3 Some Examples
242(4)
5.3 Evolutionary Variational Inequalities
246(11)
5.3.1 General Settings
247(2)
5.3.2 Comparison Principle
249(6)
5.3.3 Obstacle Problem
255(2)
5.4 Sub-Supersolutions and Monotone Penalty Approximations
257(10)
5.4.1 Hypotheses and Preliminary Results
258(2)
5.4.2 Obstacle Problem
260(2)
5.4.3 Generalized Obstacle Problem
262(5)
5.5 Systems of Variational Inequalities
267(10)
5.5.1 Notations and Assumptions
268(1)
5.5.2 Preliminaries
269(3)
5.5.3 Comparison Principle for Systems
272(2)
5.5.4 Generalization, Minimal and Maximal Solutions
274(1)
5.5.5 Weakly Coupled Systems and Extremal Solutions
275(2)
5.6 Notes and Comments
277(2)
6 Hemivariational Inequalities 279(40)
6.1 Notion of Sub-Supersolution
281(4)
6.2 Quasilinear Elliptic Hemivariational Inequalities
285(14)
6.2.1 Comparison Principle
286(4)
6.2.2 Extremal Solutions and Compactness Results
290(3)
6.2.3 Application
293(6)
6.3 Evolutionary Hcrnivariational Inequalities
299(17)
6.3.1 Sub-Supersolutions and Equivalence of Problems
301(2)
6.3.2 Existence and Comparison Results
303(7)
6.3.3 Compactness and Extremality Results
310(6)
6.4 Notes and Comments
316(3)
7 Variational-Hemivariat ional Inequalit ies 319(60)
7.1 Elliptic Variational-Hemivariational Inequalities
319(17)
7.1.1 Comparison Principle
320(8)
7.1.2 Compactness and Extremality
328(8)
7.2 Evolution Variational-Hemivariational Inequalities
336(19)
7.2.1 Definitions and Hypotheses
338(2)
7.2.2 Preliminary Results
340(3)
7.2.3 Existence and Comparison Result
343(8)
7.2.4 Compactness and Extremality
351(4)
7.3 Nonsmooth Critical Point Theory
355(7)
7.4 A Constraint Hemivariational Inequality
362(6)
7.5 Eigenvalue Problem for a Variational-Hemivariational Inequality
368(7)
7.6 Notes and Comments
375(4)
List of Symbols 379(2)
References 381(12)
Index 393