Calculus of Variations and Geometric Measure Theory

F. Prinari - E. Zappale

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

created on 27 Sep 2019
modified by prinari on 11 Sep 2020


Published Paper

Inserted: 27 sep 2019
Last Updated: 11 sep 2020

Journal: Jota
Year: 2020
Doi: 10.1007/s10957-020-01712-y


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.