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

revised version: November 2005
Adriana Garroni, Marcello Ponsiglione, and Francesca Prinari
(Please use for correspondence this email).

Submission date: 13. Jan. 2005
Pages: 23
published in: Calculus of variations and partial differential equations, 27 (2006) 4, p. 397-420 
DOI number (of the published article): 10.1007/s00526-005-0354-5
MSC-Numbers: 47J20, 58B20, 49J45
Keywords and phrases: variational methods, supremal functionals, finsler metric, relaxation, gamma convergence
Download preprint: PDF (305 kB), PS ziped (262 kB)

Abstract:
In this paper we consider positively 1-homogeneous supremal functionals of the type . We prove that the relaxation is a difference quotient, that is

where is a geodesic distance associated to F. Moreover we prove that the closure of the class of 1-homogeneous supremal functionals with respect to -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.