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

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 $Ω = Ω_{0} ∖ {\bar{B}}_{R_{1}}$ , where $B_{R_{1}}$ is the ball centered at the origin with radius $R_{1} > 0$ and $Ω_{0} \subset ℝ^{n}$ , $n \geq 2$ , is an open, bounded and convex set such that $B_{R_{1}} ⋐ Ω_{0}$ , then the first Steklov–Dirichlet eigenvalue $σ_{1} (Ω)$ has a maximum when $R_{1}$ and the measure of $Ω$ are fixed. Moreover, if $Ω_{0}$ is contained in a suitable ball, we prove that the spherical shell is the maximum.

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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

Authors

Nunzia Gavitone
Dipartimento di Matematica e Applicazioni “R. Caccioppoli” Università degli studi di Napoli Federico II Naples Italy

Gloria Paoli
Department of Data Science Friedrich-Alexander-Universität Erlangen-Nürnberg Erlangen Germany

Gianpaolo Piscitelli
Dipartimento di Matematica e Applicazioni “R. Caccioppoli” Università degli studi di Napoli Federico II Naples Italy

Rossano Sannipoli
Dipartimento di Matematica e Applicazioni “R. Caccioppoli” Università degli studi di Napoli Federico II Naples Italy

Vol. 320, No. 2, 2022

Nunzia Gavitone, Gloria Paoli, Gianpaolo Piscitelli and Rossano Sannipoli

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification

Milestones

Authors