A monotonicity result for the first Steklov-Dirichlet Laplacian eigenvalue

TL;DR

This paper proves a monotonicity property of the first Steklov-Dirichlet eigenvalue for Laplace operators in symmetric annular domains with a spherical hole, showing how the eigenvalue changes as the hole varies.

Contribution

It establishes a new monotonicity result for the first Steklov-Dirichlet eigenvalue in symmetric annular domains with a spherical hole.

Findings

Eigenvalue varies monotonically with the size of the spherical hole.

The result applies specifically to centrally symmetric outer regions.

Provides theoretical insight into spectral properties of Laplacian in complex domains.

Abstract

In this paper, we consider the first Steklov-Dirichlet eigenvalue of the Laplace operator in annular domain with a spherical hole. We prove a monotonicity result with respect the hole, when the outer region is centrally symmetrc.