From 1-homogeneous supremal functionals to difference quotients: relaxation and $Gamma$-convergence.

In this paper we consider positively $1$-homogeneous supremal functionals of the type $F(u):=\supess_\Om f(x,\nabla u(x))$. We prove that the relaxation $\bar F$ is a {\it difference quotient}, that is $$ \bar{F}(u)=R^{d_F}(u):= \sup_{x,y\in \Om,\,x\neq y} \frac{u(x) - u(y)}{d_F(x,y)} \qquad \text{ for every } u\in \wi, $$ where $d_F$ is a geodesic distance associated to $F$. Moreover we prove that the closure of the class of $1$-homogeneous supremal functionals with respect to $\Gamma$-convergence is given exactly by the class of difference quotients associated to geodesic distances. This class strictly contains supremal functionals, as the class of geodesic distances strictly contains {\it intrinsic} distances.