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

Abstract

In this paper we prove the existence of a maximum for the first Steklov-Dirichlet Laplacian eigenvalue in the class of convex sets with a fixed spherical hole under volume constraint. More precisely, if Ω " Ω 0 zB R1 , where B R1 is the ball centered at the origin with radius R 1 ą 0 and Ω 0 Ă R n , n ě 2, is an open bounded and convex set such that B R1 Ť Ω 0 , then the first Steklov-Dirichlet Laplacian eigenvalue σ 1 pΩq 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.

Steklov-Dirichlet spectrum: stability, optimization and continuity of eigenvalues

“…This kind of mixed eigenvalues has been deeply studied. For instance in [14] bounds for the Riesz mean has been obtained, in [5] the authors obtained inequalities between Steklov-Dirichlet eigenvalues and Steklov-Neumann eigenvalues, in [17] the authors proved a two terms asymptotic formula and in [12,11,23] optimization of the first Steklov-Dirichlet eigenvalue on doubly connected domains has been studied.…”

“…From the regularity assumption on the domain Ω we know that the trace operator is compact and we denote its norm by C t . Using this fact, combined with (12) we obtain the following upper bound for Γ S u 2 k :…”