An indirect method in the calculus of variations

Abstract

This article presents a new approach to the issue of the existence of solutions to the basic problem in the calculus of variations. The method is indirect, and applies to certain classes of of problems with slow or no growth, in addition to those satisfying the traditional coercivity condition. The proof hinges upon showing with the help of nonsmooth analysis that a certain value function is constant. Examples are given to illustrate the applicability of the results and the necessity of the hypotheses.