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) := sup(Omega)f(x, del u(x)). We prove that the relaxation (F) over bar is a difference quotient, that is (F) over bar (u) =R-dF (u) := sup(x, y is an element of Omega, x not equal y) (u(x) - u(y))/(dF(x, y)) for every u is an element of W-1,W-infinity (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.