Calculus of Variations and Geometric Measure Theory

B. Bogosel - A. Henrot - I. Lucardesi

Minimization of the eigenvalues of the Dirichlet-Laplacian with a diameter constraint

created by lucardesi on 08 Jan 2018
modified on 05 Oct 2018


Published Paper

Inserted: 8 jan 2018
Last Updated: 5 oct 2018

Journal: SIAM J. Math. Anal.
Volume: 50
Number: 5
Pages: 5337–5361
Year: 2018
Doi: 10.1137/17M1162147


In this paper we look for the domains minimizing the $h$-th eigenvalue of the Dirichlet-Laplacian $\lambda_h$ with a constraint on the diameter. Existence of an optimal domain is easily obtained, and is attained at a constant width body. In the case of a simple eigenvalue, we provide non standard (i.e., non local) optimality conditions. Then we address the question whether or not the disk is an optimal domain in the plane, and we give the precise list of the 17 eigenvalues for which the disk is a local minimum. We conclude by some numerical simulations showing the 20 first optimal domains in the plane.