Publisher's Synopsis
The main goal of the book is to study the lower semicontinuity, relaxation and integral representation results for several kinds of problems in the Calculus of Variations, though it does not deal with these problems in the space of functions with bounded variation.;Among the topics covered, the author looks at the topological preliminaries and the abstract setting of the Direct Method in general topological spaces and deals with the classical functionals of the Calculus of Variations and also relaxation in Optimal Control Theory.;While the book deals with problems involving a single functional, it is possible to develop a theory in which sequences of functionals are considered and the asymptotic behaviour of minimum points (instead of the existence of a minimum point) is studied. This is the theory of gamma-convergence, which is involved in all cases in which one wants to deduce the asymptotic behaviour of the solutions of a sequence of variational problems from the asymptotic behaviour of the related energy functionals.
