Existence of Minimizers for Non-Level Convex Supremal Functionals

TL;DR

This paper establishes necessary and sufficient conditions for the existence of solutions to non-level convex supremal functionals, extending the understanding of minimizers in variational problems with non-convex integrands.

Contribution

It provides a characterization of solution existence for supremal functionals without level convexity, using differential inclusions and boundary conditions.

Findings

Derived conditions linking supremal functional solutions to differential inclusions.

Compared non-level convex cases with level convex problems to establish existence criteria.

Investigated convexity conditions of the supremand function.

Abstract

The paper is devoted to determine necessary and sufficient conditions for existence of solutions to the problem ${\rm inf}{{\rm ess sup}_{x \in \Omega} f(\nabla u(x)): u \in u_0 + W^{1,\infty}_0(\Omega)}$ 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.