Existence of Minimizers for NonLevel Convex Supremal Functionals
Ana Margarida Ribeiro, Elvira Zappale · SIAM Journal on Control and Optimization · 2014
The paper is devoted to determining necessary and sufficient conditions for existence of solutions to the problem $\inf\{ {\mathop{\rm ess\: sup }}_{x\in\Omega}\, f( abla u(x)): u \in u_0 + W^{1,\infty}_0(\Omega)\}$, 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.