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arXiv · 1306.0392

Faber-Krahn inequalities in sharp quantitative form

Lorenzo Brasco·Guido De Philippis·Bozhidar Velichkov
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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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Lorenzo Brasco, Guido De Philippis, Bozhidar Velichkov. 2013-06-03. Faber-Krahn inequalities in sharp quantitative form. https://doi.org/10.1215/00127094-3120167

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