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

Nunzia Gavitone, Gloria Paoli, Gianpaolo Piscitelli and Rossano Sannipoli

Vol. 320 (2022), No. 2, 241–259

DOI: 10.2140/pjm.2022.320.241

Abstract

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 $\mathrm{\Omega }={\mathrm{\Omega }}_0\setminus {\overline{B}}_{R_1}$, where $B_{R_1}$ is the ball centered at the origin with radius $R_1>0$ and ${\mathrm{\Omega }}_0\subset {ℝ}^n$, $n\ge 2$, is an open, bounded and convex set such that $B_{R_1}⋐{\mathrm{\Omega }}_0$, then the first Steklov–Dirichlet eigenvalue ${\sigma }_1(\mathrm{\Omega })$ has a maximum when $R_1$ and the measure of $\mathrm{\Omega }$ are fixed. Moreover, if ${\mathrm{\Omega }}_0$ is contained in a suitable ball, we prove that the spherical shell is the maximum.

Keywords

Laplacian eigenvalue, Steklov–Dirichlet boundary conditions, isoperimetric inequality

Mathematical Subject Classification

Primary: 28A75, 35J25, 35P15

Milestones

Received: 23 July 2021

Revised: 4 June 2022

Accepted: 7 August 2022

Published: 15 February 2023