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Journal of Convex Analysis 14 (2007), No. 3, 515--541
Copyright Heldermann Verlag 2007

Principles of Comparison with Distance Functions for Absolute Minimizers

Thierry Champion
Lab. d'Analyse Non Linéaire Appliquée, U.F.R. des Sciences et Techniques, Université du Sud Toulon-Var, BP 20132, 83957 La Garde, France

Luigi De Pascale
Dip. di Matematica Applicata, Università di Pisa, Via Bonanno Pisano 25/B, 56126 Pisa, Italy

We extend the principle of comparison with cones introduced by M. G. Crandall, L. C. Evans and R. F. Gariepy [Calc. Var. Partial Diff. Equations 13 (2001) 123--139] for the Minimizing Lipschitz Extension Problem to a wide class of supremal functionals. This gives a geometrical characterization of the absolute minimizers (optimal solutions whose minimality is local). Some application to the stability of absolute minimizers with respect to the Γ-convergence is given. A variation of the basic idea also allows to characterize the minimal Lipschitz extensions in length metric spaces.

Keywords: Supremal functionals, absolute minimizers, comparison with cones, comparison with distance functions, minimal Lipschitz extensions.

MSC: 49K30, 65K10

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