On two functionals connected to the Laplacian in a class of doubly connected domains in space-forms

TL;DR

This work investigates two functionals linked to the Laplacian on doubly connected domains in space forms, showing that both extremize when the inner and outer balls are concentric. The authors develop a Riemannian shape calculus for the stationary and spectral Dirichlet problems, derive the corresponding material and shape derivatives, and compute the Eulerian derivatives of the functionals under two-ball perturbations. They prove that the domain-functional J is minimized and the first Dirichlet eigenvalue functional J1 is maximized precisely at concentric configurations in S^n and H^n. The results combine geometric domain variation with elliptic theory to yield sharp extremality statements for curved-space annuli.

Abstract

Let $B_1$ be a ball of radius $r_1$ in $S^n(\Hy^n)$, and let $B_0$ be a smaller ball of radius $r_0$ such that $\bar{B_0}\subset B_1$. For $S^n$ we consider $r_1< \pi$. Let $u$ be a solution of the problem $-\La u =1$ in $\Om := B_1\setminus \bar{B_0}$ vanishing on the boundary. It is shown that the associated functional $J(\Om)$ is minimal if and only if the balls are concentric. It is also shown that the first Dirichlet eigenvalue of the Laplacian on $\Om$ is maximal if and only if the balls are concentric.

Theorems & Definitions (1)

• proof