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 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 ${\Omega }_0\subset {\mathbb{R}}^n$, $n\ge 2$, is an open bounded and convex set such that $B_{R_1}⋐{\Omega }_0$, 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.

19 pages

References in corpus (3)

• Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems
• A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle
• Shape monotonicity of the first Steklov-Dirichlet eigenvalue on eccentric annuli

Cited by in corpus (5)

• Piecewise nonlinear materials and Monotonicity Principle
• The p-Laplace "Signature" for Quasilinear Inverse Problems with Large Boundary Data
• A stability result for the first Robin-Neumann eigenvalue: A double perturbation approach
• The $p_0$-Laplace "Signature" for Quasilinear Inverse Problems
• Approximation of the first Steklov-Dirichlet eigenvalue on eccentric spherical shells in general dimensions