Faber-Krahn inequalities in sharp quantitative form

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 Poincaré-Sobolev constant for the embeddings $W^{1,2}_0(Ω)\hookrightarrow L^q(Ω)$.

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