Faber-Krahn inequalities in sharp quantitative form

TL;DR

This paper provides a sharp quantitative version of the Faber-Krahn inequality, confirming a conjecture and extending the result to optimal Poincaré-Sobolev constants for Sobolev embeddings.

Contribution

It proves a sharp quantitative enhancement of the Faber-Krahn inequality, validating a conjecture and applying to Poincaré-Sobolev constants for Sobolev embeddings.

Findings

Confirmed a conjecture by Nadirashvili and Bhattacharya-Weitsman.

Established a sharp quantitative form of the Faber-Krahn inequality.

Extended results to optimal Poincaré-Sobolev constants.

Abstract

The classical Faber-Krahn inequality asserts that balls (uniquely) minimize the first eigenvalue of the Dirichlet-Laplacian among sets with given volume. In this paper we prove a sharp quantitative enhancement of this result, thus confirming a conjecture by Nadirashvili and Bhattacharya-Weitsman. More generally, the result applies to every optimal Poincar\'e-Sobolev constant for the embeddings $W^{1,2}_0(\Omega)\hookrightarrow L^q(\Omega)$.