Unbounded Functionals in the Calculus of Variations: Representation, Relaxation, and Homogenization
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Summary
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.
Table of 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
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