Abstract

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 C1C^1. 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.

Author information

Contact details are reproduced from the original publication and may be historical.

Dorin Bucur

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

bucur@math.univ-metz.fr

Keywords

  • convex sets
  • regularity
  • shape optimisation

Mathematics Subject Classification

35J20, 35B20

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Suggested citation

D. Bucur. “Regularity of Optimal Convex Shapes.” Journal of Convex Analysis 10 (2003), No. 2, 501–516. https://doi.org/10.68381/jca1031

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Published by Heldermann Verlag, 2003. Rights now held by Banach Press LLC.