On the minimization of Dirichlet eigenvalues of the Laplace operator

TL;DR

This paper investigates the minimization of Dirichlet eigenvalues of the Laplace operator under boundary measure constraints, establishing existence, convexity, connectivity, and component bounds for minimizers across various dimensions and eigenvalues.

Contribution

It provides new existence results, properties, and bounds for minimizers of Dirichlet eigenvalues with boundary measure constraints in multiple dimensions.

Findings

Existence of convex minimizers for the second and higher eigenvalues in 2D.

Minimizers' interior is also a minimizer, and their complements are connected.

Bounds on the number of components of minimizers depending on dimension and eigenvalue.

Abstract

We study the variational problem $$\inf \{\lambda_k(\Omega): \Omega\ \textup{open in}\ \R^m,\ |\Omega| < \infty, \ \h(\partial \Omega) \le 1 \},$$ where $\lambda_k(\Omega)$ is the $k$'th eigenvalue of the Dirichlet Laplacian acting in $L^2(\Omega)$, $\h(\partial \Omega)$ is the $(m-1)$- dimensional Hausdorff measure of the boundary of $\Omega$, and $|\Omega|$ is the Lebesgue measure of $\Omega$. If $m=2$, and $k=2,3, \cdots$, then there exists a convex minimiser $\Omega_{2,k}$. If $m \ge 2$, and if $\Omega_{m,k}$ is a minimiser, then $\Omega_{m,k}^*:= \textup{int}(\overline{\Omega_{m,k}})$ is also a minimiser, and $\R^m\setminus \Omega_{m,k}^*$ is connected. Upper bounds are obtained for the number of components of $\Omega_{m,k}$. It is shown that if $m\ge 3$, and $k\le m+1$ then $\Omega_{m,k}$ has at most $4$ components. Furthermore $\Omega_{m,k}$ is connected in the following cases : (i) $m\ge 2, k=2,$ (ii) $m=3,4,5,$ and $k=3,4,$ (iii) $m=4,5,$ and $k=5,$ (iv) $m=5$ and $k=6$. Finally, upper bounds on the number of components are obtained for minimisers for other constraints such as the Lebesgue measure and the torsional rigidity.