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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.

Received:
Jan 13, 2005
Published:
Jan 13, 2005
MSC Codes:
47J20, 58B20, 49J45
Keywords:
variational methods, supremal functionals, finsler metric, relaxation, gamma convergence

Related publications

inJournal
2006 Repository Open Access
Adriana Garroni, Marcello Ponsiglione and Francesca Prinari

From 1-homogeneous supremal functionals to difference quotients: relaxation and [gamma]-convergence

In: Calculus of variations and partial differential equations, 27 (2006) 4, pp. 397-420