Introduction

Abstract

The historical problem of the calculus of variations and of the theory of optimization is that of finding minimizers of functionals in the form of integrals defined over infinite dimensional spaces. Historically, these problems were tackled and in many instances solved through the associated Euler-Lagrange equation, which is the analogue of the critical point condition for functions defined over finite dimensional spaces. This condition usually leads to an equation or system of ordinary differential equations or partial differential equations. The search for minimizers (or in general extremals) was reduced in this way to finding certain solutions of differential equations associated to the corresponding functional. Whenever these solutions could be found explicitly or shown to exist, one would establish, under suitable assumptions, the existence of solutions to the variational principle. This way of proceeding is especially fruitful in one dimension, when the Euler-Lagrange equation is an ordinary differential equation or system. In higher dimensions nonlinear partial differential equations need to be solved and in general it is not an easy task to show existence of solutions. Consequently, attention was focussed on finding extremals directly from the functional itself: the direct method of the calculus of variations was the outcome. This method has been so successful that today it is one of the usual ways of showing existence of solutions to many nonlinear elliptic partial differential equations.