# A reverse quantitative isoperimetric type inequality for the Dirichlet Laplacian

### Gloria Paoli

Università degli studi di Napoli Federico II, Italy

## 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 $n≥2$, there exists a constant $c>0$, depending only on $n$, such that, for every $Ω⊂R_{n}$ open, bounded, and convex set with volume equal to the volume of a ball $B$ with radius $1$, it holds that $λ_{1}(Ω)−λ_{1}(B)≥c(P(Ω)−P(B))_{2}$, where $λ_{1}(⋅)$ denotes the first Dirichlet eigenvalue of a set and $P(⋅)$ its perimeter. The heart of the present paper is a sharp estimate of the Fraenkel asymmetry in terms of the perimeter.

## Cite this article

Gloria Paoli, A reverse quantitative isoperimetric type inequality for the Dirichlet Laplacian. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 33 (2022), no. 2, pp. 361–384

DOI 10.4171/RLM/973