# A Relaxation Result in the Vectorial Setting and Power Law Approximation for Supremal Functionals

created by zappale1 on 27 Sep 2019
modified on 18 May 2020

[BibTeX]

Submitted Paper

Inserted: 27 sep 2019
Last Updated: 18 may 2020

Year: 2019

Abstract:

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