Unbounded Functionals in the Calculus of Variations: Representation, Relaxation, and Homogenization
Luciano Carbone, Riccardo De Arcangelis · 2001
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