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

TL;DR

This paper extends relaxation results for supremal functionals in vectorial settings, establishing connections with indicator functionals and exploring $L^p$-approximation for non-lower semicontinuous cases.

Contribution

It provides a relaxation framework for supremal functionals with level convex integrands and discusses $L^p$-approximation for measurable densities, extending previous lower semicontinuity results.

Findings

Relaxation for supremal functionals with level convex integrands.

Connection established between supremal functionals and indicator functionals.

Discussion on $L^p$-approximation for non-lower semicontinuous supremal functionals.

Abstract

We provide relaxation for not lower semicontinuous supremal functionals of the type $W^{1,\infty}(\Omega;\mathbb R^d) \ni u \mapsto\supess_{ x \in \Omega}f(\nabla u(x))$ 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 $\L^N \otimes \B_{d \times N}$-measurable.