A sufficient condition for the lower semicontinuity of nonlocal supremal functionals in the vectorial case

TL;DR

This paper establishes a sufficient condition for the lower semicontinuity of certain nonlocal supremal functionals involving vector-valued functions, which is crucial for variational calculus and optimization problems.

Contribution

It introduces a new sufficient condition that guarantees lower semicontinuity for a class of nonlocal supremal functionals in the vectorial setting.

Findings

Provides a clear criterion for lower semicontinuity of nonlocal supremal functionals.

Extends the understanding of semicontinuity in the context of vector-valued functions.

Lays groundwork for further analysis and applications in nonlocal variational problems.

Abstract

In this note we present a sufficient condition ensuring lower semicontinuity for nonlocal supremal functionals of the type $$W^{1,\infty}(\Omega;\mathbb R^d)\ni u \mapsto \sup{\rm ess}_{(x,y)\in \Omega} W(x,y, \nabla u(x),\nabla u(y)),$$ where $\Omega$ is a bounded open subset of $\mathbb R^N$ and $W:\Omega \times \Omega \times \mathbb R^{d \times N}\times \mathbb R^{d \times N} \to \mathbb R$.