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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
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 Ω = Ω0 B ¯R1, where BR1 is the ball centered at the origin with radius R1 > 0 and Ω0 n, n 2, is an open, bounded and convex set such that BR1 Ω0, then the first Steklov–Dirichlet eigenvalue σ1(Ω) has a maximum when R1 and the measure of Ω are fixed. Moreover, if Ω0 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
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