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 ě 2, there exists a constant c ą 0, depending only on n, such that, for every Ω Ă R open, bounded and convex set with volume equal to the volume of a ball B with radius 1, it holds λ1pΩq ́ λ1pBq ě c pP pΩq ́ P pBqq , where by λ1p ̈q we denote the first Dirichlet eigenvalue of a set and by P p ̈q its perimeter. The hearth of the present paper is a sharp estimate of the Fraenkel asymmetry in terms of the perimeter. MSC 2020: 35J05, 35J57, 52A27