MiS Preprint Repository

In this paper we consider positively $1$-homogeneous supremal functionals of the type $F(u):=\underset{\mathrm{\Omega }}{sup}f(x,\mathrm{\nabla }u(x))$. We prove that the relaxation $\overline{F}$ is a difference quotient, that is $$\overline{F}(u)=R^{d_F}(u):=\underset{x,y\in \mathrm{\Omega },x\ne y}{sup}\frac{u(x)-u(y)}{d_F(x,y)}\text{ for every }u\in W^{i,\mathrm{\infty }}(\mathrm{\Omega }),$$ 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 $\mathrm{\Gamma }$-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 intrinsic distances.