*Published Paper*

**Inserted:** 16 may 2020

**Last Updated:** 19 nov 2021

**Journal:** Nonlinear Analysis

**Year:** 2021

**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.