Remarks on lipschitz solutions to measurable differential inclusions and an existence result for some nonconvex variational problems

Abstract

In this paper we examine the problem of finding a Lipschitz function on an open domain with prescribed boundary values and whose gradient is required to satisfy some nonhomogeneous pointwise constraints a.e. in the domain. These constraints are supposed to be given by a measurable set-valued mapping with convex, uniformly compact and nonempty-interior values. We discuss existence and metric properties of maximal solutions of such a problem. We exploit some connections with weak solutions to discontinuous Hamilton-Jacobi equations, and we provide a variational principle that characterizes maximal solutions. We investigate the case where the original problem is supplemented with bilateral obstacle constraints on the function values. Finally, as an application of these results, we prove existence for a specific class of nonconvex problems from the calculus of variations, with and without obstacle constraints, under mild regularity hypotheses on the data.