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Unbounded Functionals in the Calculus of Variations

Representation, Relaxation, and Homogenization

By Luciano Carbone, Riccardo De Arcangelis

Series Editor: Alan Jeffrey, Haim Brezis, Ronald G. Douglas

Published December 12th 2001 by Chapman and Hall/CRC – 408 pages

Series: Monographs and Surveys in Pure and Applied Mathematics

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Description

Over the last few decades, research in elastic-plastic torsion theory, electrostatic screening, and rubber-like nonlinear elastomers has pointed the way to some interesting new classes of minimum problems for energy functionals of the calculus of variations. This advanced-level monograph addresses these issues by developing the framework of a general theory of integral representation, relaxation, and homogenization for unbounded functionals.

The first part of the book builds the foundation for the general theory with concepts and tools from convex analysis, measure theory, and the theory of variational convergences. The authors then introduce some function spaces and explore some lower semicontinuity and minimization problems for energy functionals. Next, they survey some specific aspects the theory of standard functionals.

The second half of the book carefully develops a theory of unbounded, translation invariant functionals that leads to results deeper than those already known, including unique extension properties, representation as integrals of the calculus of variations, relaxation theory, and homogenization processes. In this study, some new phenomena are pointed out. The authors' approach is unified and elegant, the text well written, and the results intriguing and useful, not just in various fields of mathematics, but also in a range of applied mathematics, physics, and material science disciplines.

Contents

Preface

Basic Notations and Recalls

ELEMENTS OF CONVEX ANALYSIS

Convex Sets and Functions

Convex and Lower Semicontinuous Envelopes in Rn

Lower Semicontinuous Envelopes of Convex Envelopes

Convex Envelopes of Lower Semicontinuous Envelopes

ELEMENTS OF MEASURE AND INCREASING SET FUNCTIONS

Measures and Integrals

Basics on Lp Spaces

Derivation of Measures

Abstract Measure Theory in Topological Settings

Local Properties of Boundaries of Open Subsets of Rn

Increasing Set Functions

Increasing Set Functionals

MINIMIZATION METHODS AND VARIATIONAL CONVERGENCES

The Direct Methods in the Calculus of Variations

G-Convergence

Applications to the Calculus of

G-Convergence in Topological Vector Spaces, and of Increasing Set Functionals

Relaxation

BV AND SOBOLEV SPACES

Regularization of Measures and of Summable Functions

BV Spaces

Sobolev Spaces

Some Compactness Criteria

Periodic Sobolev Functions

LOWER SEMICONTINUITY AND MINIMIZATION OF INTEGRAL FUNCTIONALS

Functionals on BV Spaces

Functionals on Sobolev Spaces

Minimization of Integral Functionals

CLASSICAL RESULTS AND MATHEMATICAL MODELS ORIGINATING UNBOUNDED FUNCTIONALS

Classical Unique Extension Results

Classical Integral Representation Results

Classical Relaxation Results

Classical Homogenization Results

Mathematical Aspects of Some Physical Models Originating Unbounded Functionals

ABSTRACT REGULARIZATION AND JENSEN'S INEQUALITY

Integral of Functions with Values in Locally Convex Topological Vector Spaces

On the Definition of a Functional on Functions and on Their Equivalence Classes

Regularization of Functions in Locally Convex Topological Vector Subspaces of L1loc Rn

Applications to Convex Functionals on BV Spaces

UNIQUE EXTENSION RESULTS

Unique Extension Result for Inner Regular Functionals

Existence and Uniqueness Results

Unique Extension Results for Measure Like Functionals

Some Applications

A Note on Lavrentiev Phenomenon

INTEGRAL REPRESENTATION FOR UNBOUNDED FUNCTIONALS

Representation on Linear Functions

Representation on Continuously Differentiable Functions

Representation on Sobolev Spaces

Representation on BV Spaces

RELAXATION OF UNBOUNDED FUNCTIONALS

Notations and Elementary Properties of Relaxed Functionals in the Neumann Case

Relaxation of Neumann Problems: the Case of Bounded Effective Domain with Nonempty Interior

Relaxation of Neumann Problems: the Case of Bounded Effective Domain with Empty Interior

Relaxation of Neumann Problems: a First Result without Boundedness Assumptions of the Effective Domain

Relaxation of Neumann Problems: Relaxation on BV Spaces

Notations and Elementary Properties of Relaxed Functionals in the Dirichlet Case

Relaxation of Dirichlet Problems

Applications to Minimum Problems

Additional Remarks on Integral Representation on the whole Space of Lipschitz Functions

CUT-OFF FUNCTIONS AND PARTITIONS OF UNITY

Cut-off Functions

Partitions of Unity

HOMOGENIZATION OF UNBOUNDED FUNCTIONALS

Notations and Basic Results

Some Properties of G-Limits

Finiteness Conditions

Representation on Affine Functions

A Blow-up Condition

Representation Results

Applications to the Convergence of Minima and of Minimizers

Explicit Computations and Remarks on Homogenized Treloar's Energies

HOMOGENIZATION OF UNBOUNDED FUNCTIONALS WITH SPECIAL CONSTRAINTS SET

Homogenization with Fixed Constraints Set: the Case of Neumann Boundary Conditions Homogenization with Fixed Constraints Set: the Case of Dirichlet Boundary Conditions

Homogenization with Fixed Constraints Set: the Case of Mixed Boundary Conditions

Homogenization with Fixed Constraints Set: Applications to the Convergence of Minima and of Minimizers

Homogenization with Oscillating Special Constraints Set

Final Remarks.

Bibliography

Index

Name: Unbounded Functionals in the Calculus of Variations: Representation, Relaxation, and Homogenization (Hardback) – Chapman and Hall/CRC 
Description: By Luciano Carbone, Riccardo De ArcangelisSeries Editor: Alan Jeffrey, Haim Brezis, Ronald G. Douglas. Over the last few decades, research in elastic-plastic torsion theory, electrostatic screening, and rubber-like nonlinear elastomers has pointed the way to some interesting new classes of minimum problems for energy functionals of the calculus of...
Categories: Mathematical Analysis, Differential Equations, Mathematical Physics