From 1-homogeneous supremal functionals to difference quotients: relaxation and Γ-convergence

Abstract

In this paper we consider positively 1-homogeneous supremal functionals of the type \( F(u) := {\rm sup}_{\Omega}f(x,\nabla u(x))\). We prove that the relaxation $\bar{F}$ is a difference quotient, that is \( \bar{F}(u) = R^{d_F}(u): = \mathop{\rm sup}_{x,y\in\Omega,x\neq y}\frac{u(x)-u(y)}{d_F(x,y)}\quad \mbox{for every}\ u\in W^{1,\infty}(\Omega),\) 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 Γ-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.