Regularity result for a shape optimization problem under perimeter constraint

We study the problem of optimizing the eigenvalues of the Dirichlet Laplace operator under perimeter constraint. We prove that optimal sets are analytic outside a closed singular set of dimension at most $d-8$ by writing a general optimality condition in the case the optimal eigenvalue is multiple. As a consequence we find that the optimal $k\textrm{-th}$ eigenvalue is strictly smaller than the optimal $k+1\textrm{-th}$ eigenvalue. We also provide an elliptic regularity result for sets with positive and bounded weak curvature.

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This work was supported by the project ANR Optiform and by the Fondation Sciences Mathematiques de Paris.

Received 24 May 2017

Accepted 27 June 2017

Published 30 December 2019