Existence of Minimizers for Non-Level Convex Supremal Functionals

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

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