The role of intrinsic distances in the relaxation of L∞-functionals

We consider a supremal functional of the form $$F(u)= supess_{x in Omega}f(x,Du(x))$$ where $Omegasubseteq R^N$ is a regular bounded open set, $uin wi$ 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 $lambda>inf_{wi} F$, 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 ot =y} rac{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.