An isoperimetric inequality for the first Steklov-Dirichlet Laplacian eigenvalue of convex sets with a spherical hole

TL;DR

This paper establishes an isoperimetric inequality for the first Steklov-Dirichlet eigenvalue of convex sets with a spherical hole, showing the spherical shell maximizes this eigenvalue under volume constraints.

Contribution

It proves the existence of a maximum eigenvalue for convex sets with a spherical hole and identifies the spherical shell as the maximizer under certain conditions.

Findings

The first Steklov-Dirichlet eigenvalue attains a maximum for convex sets with a fixed spherical hole.

The spherical shell is the maximizer of the eigenvalue when the convex set is contained in a suitable ball.

Existence of a maximum eigenvalue under volume constraint is established.

Abstract

In this paper we prove the existence of a maximum for the first Steklov-Dirichlet eigenvalue in the class of convex sets with a fixed spherical hole under volume constraint. More precisely, if $\Omega=\Omega_0 \setminus \bar{B}_{R_1}$, where $B_{R_1}$ is the ball centered at the origin with radius $R_1>0$ and $\Omega_0\subset\mathbb{R}^n$, $n\geq 2$, is an open bounded and convex set such that $B_{R_1}\Subset \Omega_0$, then the first Steklov-Dirichlet eigenvalue $\sigma_1(\Omega)$ has a maximum when $R_1$ and the measure of $\Omega$ are fixed. Moreover, if $\Omega_0$ is contained in a suitable ball, we prove that the spherical shell is the maximum.