The role of intrinsic distances in the relaxation of $L^\infty$-functionals

TL;DR

This paper characterizes the relaxation of certain supremal functionals in terms of intrinsic distances, showing convexity of sublevel sets and linking relaxed functionals to difference quotient functionals.

Contribution

It provides a new description of the relaxed form of supremal functionals using intrinsic distances and difference quotient functionals, extending understanding of their lower semicontinuous envelopes.

Findings

Relaxed functional coincides with the difference quotient functional $R_{d^1_F}$ for positive 1-homogeneous cases.

Sublevel sets of relaxed supremal functionals are convex under various topologies.

Intrinsic distances are key to describing the relaxation of supremal functionals.

Abstract

We consider a supremal functional of the form $$F(u)=\mathop{\rm ess\: sup }_{x \in \Omega} f(x,Du(x))$$ where $\Omega\subseteq \mathbf {R}^N$ is a regular bounded open set, $u\in W^{1,\infty}(\Omega)$ and $f$ is a Borel function. Assuming that the intrinsic distances $d^{\lambda}_F(x,y):= \sup \Big\{ u(x) - u(y): \, F(u)\leq \lambda \Big\}$ are locally equivalent to the euclidean one for every $\lambda>\inf_{W^{1,\infty}(\Omega)} F$, we give a description of the sublevel sets of the weak$^*$-lower semicontinuous envelope of $F$ in terms of the sub-level sets of the difference quotient functionals $R_{d^\lambda_F}(u):=\sup_{x\not =y} \frac{u(x)-u(y)}{d^\lambda_F(x,y)}. $ As a consequence we prove that the relaxed functional of positive $1$-homogeneous supremal functionals coincides with $R_{d^1_F}$. Moreover, for a more general supremal functional $F$ (a priori non coercive), we prove that the sublevel sets of its relaxed functionals with respect to the weak$^*$ topology, the weak$^*$ convergence and the uniform convergence are convex. The proof of these results relies both on a deep analysis of the intrinsic distances associated to $F$ and on a careful use of variational tools such as $\Gamma$-convergence.