A reverse quantitative isoperimetric type inequality for the Dirichlet Laplacian

Abstract

A stability result in terms of the perimeter is obtained for the first Dirichlet eigenvalue of the Laplacian operator. In particular, we prove that, once we fix the dimension $n\ge 2$, there exists a constant $c>0$, depending only on $n$, such that, for every $\Omega \subset {\mathbb{R}}^n$ open, bounded, and convex set with volume equal to the volume of a ball $B$ with radius $1$, it holds that ${\lambda }_1(\Omega )-{\lambda }_1(B)\ge c(P(\Omega )-P(B){)}^2$, where ${\lambda }_1(\cdot )$ denotes the first Dirichlet eigenvalue of a set and $P(\cdot )$ its perimeter. The heart of the present paper is a sharp estimate of the Fraenkel asymmetry in terms of the perimeter.