Regularity result for a shape optimization problem under perimeter constraint

Beniamin Bogosel

Published 2016-06-19Version 1

We study the problem of optimizing the eigenvalues of the Dirichlet Laplace operator under perimeter constraint. We prove that optimal sets are smooth 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.