On eigenvalue problems related to the laplacian in a class of doubly connected domains

Abstract

We study eigenvalue problems in some specific class of doubly connected domains. In particular, we prove the following.

• 1. Let $B_1$ be an open ball in ${\mathbb {R}}^n$, $n>2$ and $B_0$ be an open ball contained in $B_1$. Then the first eigenvalue of the problem

  $$\begin{aligned} \begin{array}{rcll} \varDelta u &{}=&{} 0 \, &{} \text{ in } \, B_1 \setminus {\overline{B}}_0 , \\ u &{}=&{} 0 \, &{} \text{ on } \, {\partial B_0}, \\ \frac{\partial u}{\partial \nu } &{}=&{} \tau u \, &{} \text{ on } \, {\partial B_1}, \end{array} \end{aligned}$$

  attains its maximum if and only if $B_0$ and $B_1$ are concentric. Here $\nu$ is the outward unit normal on $\partial B_1$ and $\tau$ is a real number.

• 2. Let $B_0 \subset M$ be a geodesic ball of radius r centered at a point $p \in M$, where M denote either a non-compact rank-1 symmetric space $({\mathbb {M}}, ds^2)$ with curvature $-4 \le K_{{\mathbb {M}}} \le -1$ or $M = {\mathbb {R}}^{m}$. Let $D \subset M$ be a domain of fixed volume which is geodesically symmetric with respect to the point $p\in M$ such that $\overline{{B}_0}\subset D$. Then the first non-zero eigenvalue of

  $$\begin{aligned} \begin{array}{rcll} \varDelta u &{}=&{} \mu u \, &{} \text{ in } \, D \setminus {\overline{B}}_0, \\ \frac{\partial u}{\partial \nu } &{}=&{} 0 \, &{} \text{ on } \, {\partial (D \setminus {\overline{B}}_0)}, \end{array} \end{aligned}$$

  attains its maximum if and only if D is a geodesic ball centered at p. Here $\nu$ represents the outward unit normal on ${\partial (D \setminus {\overline{B}}_0)}$ and $\mu$ is a real number.