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 article we prove a sharp quantitative enhancement of this result, thus confirming a conjecture by Nadirashvili and by Bhattacharya and Weitsman. More generally, the result applies to every optimal Poincaré-Sobolev constant for the embeddings W 0 1,2 (ω) {right arrow, hooked} Lq(ω).