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

Abstract

We consider a supremal functional of the form $$F(u)={\mathrm{esssup}}_{x\in \Omega }f(x,Du(x))$$ 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 $d_F^{\lambda }(x,y):=\sup \{u(x)-u(y):F(u)\le \lambda \}$ 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${}^{\ast }$-lower semicontinuous envelope of $F$ in terms of the sub-level sets of the difference quotient functionals $R_{d_F^{\lambda }}(u):={\sup }_{x\ne y}\frac{u(x)-u(y)}{d_F^{\lambda }(x,y)}.$ As a consequence we prove that the relaxed functional of positive $1$-homogeneous supremal functionals coincides with $R_{d_F^1}$. 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${}^{\ast }$ topology, the weak${}^{\ast }$ 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.