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Journal of Convex Analysis 10 (2003), No. 2, 501--516
Copyright Heldermann Verlag 2003

Regularity of Optimal Convex Shapes

Dorin Bucur
Dép. de Mathématiques, UMR-CNRS 7122, Université de Metz, Ile du Saulcy, 57045 Metz, France,

We consider shape optimisation problems in the class of convex sets. Assuming that the shape functional satisfies a Lipschitz like property with respect to a distance issued from the γ-convergence, we prove that the minimiser has the boundary of class C1. In particular, we prove that large classes of functionals depending on the eigenvalues of the Dirichlet Laplacian satisfy this property. The key point of the paper is the understanding of the asymptotic behaviour of the γ-convergence near the "angular" points of the convex set.

Keywords: convex sets, regularity, shape optimisation.

MSC 1991: 35J20, 35B20.

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