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Journal of Convex Analysis 11 (2004), No. 1, 017--040
Copyright Heldermann Verlag 2004

A Duality Approach for Variational Problems in Domains with Cracks

François Ebobisse
Dept. of Mathematics, University of Cape Town, Rondebosch 7700, South Africa,

Marcello Ponsiglione
S.I.S.S.A., Via Beirut 2-4, 34014 Trieste, Italy,

We study the asymptotic behaviour of the solutions of some minimization problems for integral functionals with convex integrands, in two-dimensional domains with cracks, under perturbations of the cracks in the Hausdorff metric. In the first part of the paper, we examine conditions for the stability of the minimum problem via duality arguments in convex optimization. In the second part, we study the limit problem in some special cases when there is no stability, using the tool of Gamma-convergence.

Keywords: capacity, convex optimization, Gamma-convergence, Hausdorff distance, integral functional.

MSC 2000: 31A15, 46N10, 49M29, 49J45.

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