Multiple integrals of Lipschitz functions in the calculus of variations

Abstract

We consider a multiple integral problem in the calculus of variations in which the integrand is locally Lipschitz but not differentiable, and in which minimization takes place over a Sobolev space. Using a minimax theorem, we derive an analogue of the classical Euler condition for optimality, couched in terms of “generalized gradients". We proceed to indicate how these results may be applied to deduce existence and smoothness properties of solutions to certain Poisson equations.