On the lower semicontinuity of supremal functionals

Gori, Michele; Maggi, Francesco

ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 135-143.

In this paper we study the lower semicontinuity problem for a supremal functional of the form $F(u,\Omega )=\underset{x\in \Omega }{\mathrm{ess}\sup }f(x,u\left(x\right),Du\left(x\right))$ with respect to the strong convergence in $L^{\infty }\left(\Omega \right)$, furnishing a comparison with the analogous theory developed by Serrin for integrals. A sort of Mazur’s lemma for gradients of uniformly converging sequences is proved.

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