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Regularity result for a shape optimization problem under perimeter constraint

Abstract : 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-th eigenvalue is strictly smaller than the optimal (k + 1)-th eigenvalue. We also provide an elliptic regularity result for sets with positive and bounded weak curvature.
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https://hal.archives-ouvertes.fr/hal-03181142
Contributor : Beniamin Bogosel <>
Submitted on : Thursday, March 25, 2021 - 2:16:13 PM
Last modification on : Sunday, March 28, 2021 - 3:25:13 AM

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Beniamin Bogosel. Regularity result for a shape optimization problem under perimeter constraint. Communications in Analysis and Geometry, 2019, 27 (7), pp.1523-1547. ⟨10.4310/CAG.2019.v27.n7.a3⟩. ⟨hal-03181142⟩

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