Communications in Analysis and Geometry

Volume 27 (2019)

Number 7

Regularity result for a shape optimization problem under perimeter constraint

Pages: 1523 – 1547



Beniamin Bogosel (Laboratoire Jean Kuntzmann, Grenoble, France)


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.

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