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