Calculus of Variations and Geometric Measure Theory

N. Gavitone - G. Paoli - G. Piscitelli - R. Sannipoli

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

created by piscitelli on 29 Sep 2022



Inserted: 29 sep 2022

Year: 2021

ArXiv: 2103.05980 PDF


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 $\Omega_0\subset\mathbb{R}^n$, $n\geq 2$, is an open bounded and convex set such that $B_{R_1}\Subset \Omega_0$, 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.