A reverse quantitative isoperimetric type inequality for the Dirichlet Laplacian

TL;DR

This paper establishes a stability inequality linking the first Dirichlet eigenvalue of a convex set to its perimeter, providing a quantitative measure of how geometric deviations affect spectral properties.

Contribution

It proves a reverse isoperimetric inequality for the Dirichlet Laplacian, relating eigenvalue differences to perimeter differences with a quadratic dependence.

Findings

Proves a lower bound for eigenvalue differences in terms of perimeter differences.

Provides a sharp estimate of Fraenkel asymmetry based on perimeter.

Establishes a constant depending only on the dimension for the inequality.

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 $n\geq2$, 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 \begin{equation*} \lambda_1(\Omega)-\lambda_1(B)\geq c\left(P(\Omega)-P(B) \right)^{2}, \end{equation*} where by $\lambda_1(\cdot)$ we denote the first Dirichlet eigenvalue of a set and by $P(\cdot)$ its perimeter. The hearth of the present paper is a sharp estimate of the Fraenkel asymmetry in terms of the perimeter.