On the minimization of Dirichlet eigenvalues

TL;DR

This paper investigates the minimization of Dirichlet eigenvalues under various geometric constraints, providing new results for open convex and quasi-open sets with fixed measures, perimeters, or other set functions.

Contribution

It introduces and analyzes two new minimization problems involving Dirichlet eigenvalues with constraints on perimeter, measure, and other set functions, extending previous spectral optimization results.

Findings

Derived bounds for eigenvalues under perimeter constraints

Established existence of minimizers in convex and quasi-open classes

Extended classical isoperimetric inequalities to eigenvalue minimization

Abstract

Results are obtained for two minimization problems: $$I_k(c)=\inf \{\lambda_k(\Omega): \Omega\ \textup{open, convex in}\ \mathbb{R}^m,\ \mathcal{T}(\Omega)= c \},$$ and $$J_k(c)=\inf\{\lambda_k(\Omega): \Omega\ \textup{quasi-open in}\ \mathbb{R}^m, |\Omega|\le 1, \mathcal {P}(\Omega)\le c \},$$ where $c>0$, $\lambda_k(\Omega)$ is the $k$'th eigenvalue of the Dirichlet Laplacian acting in $L^2(\Omega)$, $|\Omega|$ denotes the Lebesgue measure of $\Omega$, $\mathcal{P}(\Omega)$ denotes the perimeter of $\Omega$, and where $\mathcal{T}$ is in a suitable class set functions. The latter include for example the perimeter of $\Omega$, and the moment of inertia of $\Omega$ with respect to its center of mass.