Variational and Monotonicity Methods in Nonsmooth Analysis (Frontiers in Mathematics)

✍ Scribed by Nicuşor Costea, Alexandru Kristály, Csaba Varga

• Publisher: Birkhäuser
• Year: 2021
• Tongue: English
• Leaves: 450
• Category: Library

✦ Synopsis

This book provides a modern and comprehensive presentation of a wide variety of problems arising in nonlinear analysis, game theory, engineering, mathematical physics and contact mechanics. It includes recent achievements and puts them into the context of the existing literature.

The volume is organized in four parts. Part I contains fundamental mathematical results concerning convex and locally Lipschits functions. Together with the Appendices, this foundational part establishes the self-contained character of the text. As the title suggests, in the following sections, both variational and topological methods are developed based on critical and fixed point results for nonsmooth functions. The authors employ these methods to handle the exemplary problems from game theory and engineering that are investigated in Part II, respectively Part III. Part IV is devoted to applications in contact mechanics.

The book will be of interest to PhD students and researchers in applied mathematics as well as specialists working in nonsmooth analysis and engineering.

✦ Table of Contents

Preface
Acknowledgments
Contents
Acronyms
Part I Mathematical Background
1 Convex and Lower Semicontinuous Functionals
1.1 Basic Properties
1.2 Conjugate Convex Functions and Subdifferentials
1.3 The Direct Method in the Calculus of Variations
1.4 Ekeland's Variational Principle
References
2 Locally Lipschitz Functionals
2.1 The Generalized Derivative and the Clarke Subdifferential
2.2 Nonsmooth Calculus on Manifolds
2.3 Subdifferentiability of Integral Functionals
References
3 Critical Points, Compactness Conditions and Symmetric Criticality
3.1 Locally Lipschitz Functionals
3.2 Szulkin Functionals
3.3 Motreanu-Panagiotopoulos Functionals
3.4 Principle of Symmetric Criticality
References
Part II Variational Techniques in Nonsmooth Analysis and Applications
4 Deformation Results
4.1 Deformations Using a Cerami-Type Compactness Condition
4.2 Deformations with Compactness Condition of Ghoussoub-PreissType
4.3 Deformations Without a Compactness Condition
4.4 A Deformation Lemma for Szulkin Functionals
References
5 Minimax and Multiplicity Results
5.1 Minimax Results with Weakened Compactness Condition
5.2 A General Minimax Principle: The ``Zero Altitude'' Case
5.3 Z2-Symmetric Mountain Pass Theorem
5.4 Bounded Saddle Point Methods for Locally Lipschitz Functionals
5.5 Minimax Results for Szulkin Functionals
5.6 Ricceri-Type Multiplicity Results for Locally Lipschitz Functions
References
6 Existence and Multiplicity Results for Differential Inclusions on Bounded Domains
6.1 Boundary Value Problems with Discontinuous Nonlinearities
6.2 Parametric Problems with Locally Lipschitz Energy Functional
6.3 Multiplicity Alternative for Parametric Differential Inclusions Driven by the p-Laplacian
6.4 Differential Inclusions Involving the p(·)-Laplacian and Steklov-Type Boundary Conditions
6.5 Dirichlet Differential Inclusions Driven by the -Laplacian
6.5.1 Variational Setting and Existence Results
6.5.2 Dropping the Ambrosetti-Rabinowitz Type Condition
6.6 Differential Inclusions Involving Oscillatory Terms
6.6.1 Localization: A Generic Result
6.6.2 Oscillation Near the Origin
6.6.3 Oscillation at Infinity
References
7 Hemivariational Inequalities and Differential Inclusions on Unbounded Domains
7.1 Hemivariational Inequalities Involving the Duality Mapping
7.2 Hemivariational Inequalities in RN
7.2.1 Existence and Multiplicity Results
7.2.2 Applications
7.3 Hemivariational Inequalities in Ω=ωRl, l≥2
7.4 Variational Inequalities in Ω=ωR
7.5 Differential Inclusions in RN
References
Part III Topological Methods for Variational and Hemivariational Inequalities
8 Fixed Point Approach
8.1 A Set-Valued Approach to Hemivariational Inequalities
8.2 Variational-Hemivariational Inequalities with Lack of Compactness
8.3 Nonlinear Hemivariational Inequalities
8.4 Systems of Nonlinear Hemivariational Inequalities
References
9 Nonsmooth Nash Equilibria on Smooth Manifolds
9.1 Nash Equilibria on Curved Spaces
9.2 Comparison of Nash-Type Equilibria on Manifolds
9.3 Nash-Stampacchia Equilibria on Hadamard Manifolds
9.3.1 Fixed Point Characterization of Nash-Stampacchia Equilibria
9.3.2 Nash-Stampacchia Equilibrium Points: Compact Case
9.3.3 Nash-Stampacchia Equilibrium Points: Non-compact Case
9.3.4 Curvature Rigidity
9.4 Examples of Nash Equilibria on Curved Settings
References
10 Inequality Problems Governed by Set-valued Maps of Monotone Type
10.1 Variational-Hemivariational Inequalities
10.2 Quasi-Hemivariational Inequalities
10.3 Variational-Like Inequalities
References
Part IV Applications to Nonsmooth Mechanics
11 Antiplane Shear Deformation of Elastic Cylinders in Contact with a Rigid Foundation
11.1 The Antiplane Model and Formulation of the Problem
11.2 Weak Formulation and Solvability of the Problem
11.3 Examples of Constitutive Laws
11.4 Examples of Friction Laws
References
12 Weak Solvability of Frictional Problems for Piezoelectric Bodies in Contact with a Conductive Foundation
12.1 The Model
12.2 Variational Formulation and Existence of Weak Solutions
References
13 The Bipotential Method for Contact Models with Nonmonotone Boundary Conditions
13.1 The Mechanical Model and Its Variational Formulation
13.2 The Connection with Classical Variational Formulations
13.3 Weak Solvability of the Model
References
A Functional Analysis
A.1 The Hahn-Banach Theorems
A.2 Weak Topologies
A.3 Reflexive Spaces
B Set-Valued Analysis
B.1 Kuratowski Convergence
B.2 Set-Valued Maps
B.3 Continuity of Set-Valued Maps
B.4 Monotonicity of Set-Valued Operators
C Geometry of Banach Spaces
C.1 Smooth Banach Spaces
C.2 Uniform Convexity, Strict Convexity and Reflexivity
C.3 Duality Mappings
D KKM-Type Theorems, Fixed Point Results and Minimax Principles
D.1 Variants of the KKM Lemma and Fixed Point Results
D.2 Minimax Results
E Linking Sets
References
Index