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. Elements of Measure and Increasing Set Functions . Minimization Methods and Variational Convergences. Bv and Sobolev Spaces. Lower Semicontinuity and Minimization of Integral Functionals. Classical Results and Mathematical Models . Abstract Regularization and Jensen's Inequality. Unique Extension Results. Integral Representation for Unbounded Functionals. Relaxation of Unbounded Functionals. Cut-off Functions and Partitions of Unity. Homogenization of Unbounded Functionals. Homogenization of Unbounded Functionals with Special Constraints Set. Bibliography. Index.
Carbone, Luciano; De Arcangelis, Riccardo