Existence of Minimizers for Non-Level Convex Supremal Functionals

Abstract

The paper is devoted to determine necessary and sufficient conditions for existence of solutions to the problem $\inf {\mathrm{esssup}}_{x\in \Omega }f(\nabla u(x)):u\in u_0+W_0^{1,\infty }(\Omega )$ when the supremand $f$ is not necessarily level convex. These conditions are obtained through a comparison with the related level convex problem and are written in terms of a differential inclusion involving the boundary datum. Several conditions of convexity for the supremand $f$ are also investigated.