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

Adriana Garroni, Marcello Ponsiglione, and Francesca Prinari

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Submission date: 13. Jan. 2005 (revised version: November 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
Bibtex
MSC-Numbers: 47J20, 58B20, 49J45
Keywords and phrases: variational methods, supremal functionals, finsler metric, relaxation, gamma convergence
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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.