Calculus of Variations and Geometric Measure Theory

A. Garroni - M. Ponsiglione - F. Prinari

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

created by ponsiglio on 02 Dec 2004
modified by garroni on 27 Dec 2006


Published Paper

Inserted: 2 dec 2004
Last Updated: 27 dec 2006

Journal: Calc. Var.
Volume: 27
Pages: 397-420
Year: 2006


In this paper we consider positively $1$-homogeneous supremal functionals of the type $$F(u):=\sup f(x,D u(x)). $$ We prove that the relaxation of $F$ is a {\it difference quotient}, that is $$ \bar{F}(u)=R{dF}(u):= \sup (u(x) - u(y))(dF(x,y)), $$ 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.

Keywords: relaxation, Gamma-convergence, Variational methods, Supremal functionals, Finsler metrics