Minimization of the eigenvalues of the dirichlet-laplacian with a diameter constraint

TL;DR

This paper investigates the problem of minimizing the h-th eigenvalue of the Dirichlet-Laplacian under a diameter constraint, identifying optimal shapes and conditions, and exploring whether the disk is optimal.

Contribution

It establishes existence of optimal domains, derives non-standard optimality conditions, and characterizes when the disk is a local minimum among planar domains.

Findings

Optimal domains exist and are bodies of constant width.

The disk is a local minimum for 17 specific eigenvalues.

Numerical simulations illustrate the shape of the first 20 optimal domains.

Abstract

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.