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MiS Preprint
6/2005
From 1-homogeneous supremal functionals to difference quotients: relaxation and $\Gamma$-convergence
Adriana Garroni, Marcello Ponsiglione and Francesca Prinari
Abstract
In this paper we consider positively $1$-homogeneous supremal functionals of the type $F(u):=\sup_\Omega f(x,\nabla u(x))$. We prove that the relaxation $\bar F$ is a difference quotient, that is $$ \bar{F}(u)=R^{d_F}(u):= \sup_{x,y\in \Omega,\,x\neq y} \frac{u(x) - u(y)}{d_F(x,y)} \qquad \text{ for every } u\in W^{i,\infty} (\Omega), $$ 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 $\Gamma$-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.
