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).
Faber-Krahn inequalities in sharp quantitative form / Brasco, L.; De Philippis, G.; Velichkov, B.. - In: DUKE MATHEMATICAL JOURNAL. - ISSN 0012-7094. - 164:9(2015), pp. 1777-1831. [10.1215/00127094-3120167]
Faber-Krahn inequalities in sharp quantitative form
Brasco L.;De Philippis G.;Velichkov B.
2015
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 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).| File | Dimensione | Formato | |
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