Inequalities for the minimal eigenvalue of the Laplacian in an annulus

TL;DR

This paper proves that the minimal Dirichlet eigenvalue of the Laplacian in an annulus decreases as the inner disc shifts away from the center, with the maximum at concentric discs.

Contribution

It establishes a monotonicity property of the minimal eigenvalue with respect to the displacement of the inner disc in an annulus.

Findings

Minimal eigenvalue decreases with displacement of the inner disc

Maximum eigenvalue occurs at concentric discs

Eigenvalue behavior characterized for annulus geometry

Abstract

It is proved that the minimal Dirichlet eigenvalue of the Laplacian in an annulus is a monotonically decreasing function of the displacement of the center of the smaller disc. The maximal value of the minimal eigenvalue is attained when the annulus is formed by two concentric discs.