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Faber–Krahn inequalities in sharp quantitative form

Brasco, Lorenzo; De Philippis, Guido; Velichkov, Bozhidar (2015), Faber–Krahn inequalities in sharp quantitative form, Duke Mathematical Journal, 164, 9, p. 1777-1831. 10.1215/00127094-3120167

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Type
Article accepté pour publication ou publié
Date
2015
Journal name
Duke Mathematical Journal
Volume
164
Number
9
Publisher
Duke University Press
Pages
1777-1831
Publication identifier
Metadata
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Author(s)
Brasco, Lorenzo
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
De Philippis, Guido
Unité de Mathématiques Pures et Appliquées [UMPA-ENSL]
Velichkov, Bozhidar cc
Laboratoire Jean Kuntzmann [LJK]
Abstract (EN)
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 W1,20(Ω)↪Lq(Ω).
Subjects / Keywords
Stability for eigenvalues; regularity for free boundaries; torsional rigidity