Geometrical Methods in Variational Problems

This chapter is of auxiliary character. We present here necessary facts from functional analysis that are used in our book. The results presented in this chapter are given without proof, since most of them are well known and have became classical.

In this chapter, we present homotopic (or deformation) methods for studying various classes of variational problems; these are abstract variational problems, problems of the classical calculus of variations, higher-dimensional variational problems, and mathematical programming problems. Conceptually, the homotopic method is based on the following observation: if in the process of deformation of a variational problem, an extremal is uniformly isolated with respect to a parameter, then its property to be a point of minimum is a homotopy invariant. This chapter is devoted to the verification of this principle, which has many applications.

The results presented in this chapter are based on the concepts of degree theory of mapping and the theory of rotation of vector fields, which is equivalent to it; these theories originate in the classical studies of Poincaré, Brouwer, Kronecker, Hopf, Leray, and Schauder. The apparatus of the degree theory of mapping is one of the basic tools of nonlinear analysis and its applications. Therefore, we present the auxiliary material of this chapter in a detailed and self-contained manner.

This chapter is devoted to certain applications of the methods that were developed in the previous chapters. Although the range of these methods is very wide, the applied component of geometrical methods is related only to the simplest and most characteristic (to the authors’ opinion) objects.