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Faber-Krahn inequalities in sharp quantitative form

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arxiv 1306.0392 v1 pith:2GFQNRSG submitted 2013-06-03 math.AP

Lorenzo Brasco , Guido De Philippis , Bozhidar Velichkov This is my paper
classification math.AP
keywords faber-krahnomegaquantitativeresultsharpappliesassertsballs
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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 $W^{1,2}_0(\Omega)\hookrightarrow L^q(\Omega)$.

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