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

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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.show more

Product details

  • Hardback | 408 pages
  • 163.1 x 241.8 x 27.9mm | 703.08g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 5 black & white illustrations
  • 1584882352
  • 9781584882350

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 Indexshow more