A relaxation result in the vectorial setting and $L^p$-approximation for $L^{\infty }$-functionals

Abstract

We provide relaxation for not lower semicontinuous supremal functionals of the type $W^{1,\infty}(Ω;\mathbb R^d) \ni u \mapsto\supess_{ x \in Ω}f(\nabla u(x))$ in the vectorial case, where $\Omega \subset {\mathbb{R}}^N$ is a Lipschitz, bounded open set, and $f$ is level convex. The connection with indicator functionals is also enlightened, thus extending previous lower semicontinuity results in that framework. Finally we discuss the $L^p$-approximation of supremal functionals, with non-negative, coercive densities $f=f(x,\xi )$, which are only $Ł^N \otimes \B_{d \times N}$-measurable.

27 pages