On Eigenvalue Problems Related to the Laplacian in a Class of Doubly Connected Domains

TL;DR

This paper investigates eigenvalue problems related to the Laplacian in doubly connected domains, establishing conditions under which certain eigenvalues are maximized, with results applicable to Euclidean and symmetric space geometries.

Contribution

It proves that the first eigenvalue attains its maximum when the inner and outer domains are concentric or symmetric, extending classical results to more general geometric settings.

Findings

Maximum eigenvalues occur for concentric or symmetric domains.

Eigenvalue optimization characterizes geometric symmetry.

Results apply to Euclidean and non-compact symmetric spaces.

Abstract

We consider two eigenvalue problems for Laplacian on some specific doubly connected domain. In particular, we study the following two eigenvalue problems. Let $B_1$ be an open ball in $\mathbb{R}^n$ and $B_0$ be a ball contained in $B_1$. Let $\nu$ be the outward unit normal on $\partial B_1$. Then the first eigenvalue of the problem \begin{align*} \begin{array}{rcll} \Delta u &=& 0 \, &\mbox{ in } \, B_1 \setminus \bar{B}_0 , \\ u &=& 0 \, &\mbox{ on } \, {\partial B_0}, \\ \frac{\partial u}{\partial \nu} &=& \tau \, u \, &\mbox{ on } \, {\partial B_1}, \end{array} \end{align*} attains maximum if and only if $B_0$ and $B_1$ are concentric. Let $D$ be a domain in a non-compact rank-$1$ symmetric space $(\mathbb{M}, ds^2)$, geodesically symmetric with respect to the point $ p\in \mathbb{M}$. Let $B_0$ be a ball in $\mathbb{M}$ centered at $p$ such that $\bar{{B}_0}\subset D$ and $\nu$ be the outward unit normal on ${\partial (D \setminus \bar{B}_0)}$. Then the first non-zero eigenvalue of \begin{align*} \begin{array}{rcll} \Delta u &=& \mu \ u \, &\mbox{ in } \, D \setminus \bar{B}_0, \\ \frac{\partial u}{\partial \nu} &=& 0 \, &\mbox{ on } \, {\partial (D \setminus \bar{B}_0)}, \end{array} \end{align*} attains maximum if and only if $D$ is a geodesic ball centered at $p$.