Relaxation and gamma-convergence of supremal functionals

Prinari, Francesca

Bollettino della Unione matematica italiana, Série 8, 9B (2006) no. 1, pp. 101-132

Résumé

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