Eigenvalues of Euclidean wedge domains in higher dimensions

TL;DR

This paper establishes a sharp lower bound for the first Dirichlet eigenvalue of the Laplacian in higher-dimensional Euclidean wedge domains using a weighted isoperimetric inequality, extending previous two-dimensional results.

Contribution

It generalizes Payne and Weinberger's two-dimensional eigenvalue bound to higher dimensions for Euclidean wedge domains using a novel isoperimetric approach.

Findings

The lower bound is sharp and only sectors attain equality.

The method extends classical bounds to higher dimensions.

The approach involves a weighted isoperimetric inequality.

Abstract

In this paper, we use a weighted isoperimetric inequality to give a lower bound on the first Dirichlet eigenvalue of the Laplacian on a bounded domain inside a Euclidean cone. Our bound is sharp, in that only sectors realize it. This result generalizes a lower bound of Payne and Weinberger in two dimensions.