The Dirichlet Integral in Sobolev Spaces

Abstract

In this chapter we deal with the problem of minimizing the Dirichlet integral in classes of Sobolev maps between two Riemannian manifolds. As it is well- known there is a tremendous literature on the subject, concerning both analytic and geometric aspects, and probably an entire monograph would not suffice to give an account of it. Here we do not aim to completeness nor to generality in stating the results. In fact we shall only discuss some analytic questions concerning mainly the so-called regularity problem; often we shall omit proofs or give them in special cases. We would only like to provide a number of information which are relevant for the sequel.