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We prove that the $\mathrm{\Gamma }$-limit in $L_{\mu }^{\mathrm{\infty }}$ of a sequence of supremal functionals of the form $F_k(u)={\mu -\mathrm{ess}\sup }_{\mathrm{\Omega }}{f}_k(x,u)$ is itself a supremal functional. We show by a counterexample that, in general, the function which represents the $\mathrm{\Gamma }$-lim $F(\cdot ,B)$ of a sequence of functionals $F_k(u,B)={\mu -\mathrm{ess}\sup }_B{f}_k(x,u)$ can depend on the set $B$ and wegive a necessary and sufficient condition to represent $F$ in the supremal form$F(u,B)={\mu -\mathrm{ess}\sup }_{B}f(x,u)$. As a corollary, if $f$ represents a supremal functional, then the level convex envelope of $f$ represents its weak* lower semicontinuous envelope.

In this paper, we prove that the $L^p$ approximants naturally associated to a supremal functional $\Gamma$-converge to it. This yields a lower semicontinuity result for supremal functionals whose supremand satisfy weak coercivity assumptions as well as a generalized Jensen inequality. The existence of minimizers for variational problems involving such functionals (together with a Dirichlet condition) then easily follows. In the scalar case we show the existence of at least one absolute minimizer (i.e. local...

In this paper, we prove that the approximants naturally associated to a supremal functional -converge to it. This yields a lower semicontinuity result for supremal functionals whose supremand satisfy weak coercivity assumptions as well as a generalized Jensen inequality. The existence of minimizers for variational problems involving such functionals (together with a Dirichlet condition) then easily follows. In the scalar case we show the existence of at least one absolute minimizer ( local solution)...