Eigenvalues of Dirichlet Laplacian within the class of open sets with constant diameter

TL;DR

This paper investigates the shape optimization of Dirichlet Laplacian eigenvalues within constant width sets, showing that the disk is not a local minimizer for most eigenvalues, revealing nuanced geometric spectral properties.

Contribution

It demonstrates that the disk is not a local minimizer for the Dirichlet Laplacian eigenvalues in the class of constant width sets, except for a few eigenvalues.

Findings

The disk is not a local minimizer for most eigenvalues.

Certain eigenvalues have the disk as a local minimizer.

The study advances understanding of spectral optimization in geometric analysis.

Abstract

This paper is about a shape optimization problem related to the Dirichlet Laplacian eingevalues in the Euclidean plane. More precisely we study the shape of the minimizer in the class of open sets of constant width. We prove that the disk is not a local minimizer except for a limited number of eigenvalues.