Duke Mathematical Journal

Faber–Krahn inequalities in sharp quantitative form

Lorenzo Brasco, Guido De Philippis, and 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 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 ) L q ( Ω ) .

Article information

Source
Duke Math. J., Volume 164, Number 9 (2015), 1777-1831.

Dates
Received: 14 June 2013
Revised: 21 September 2014
First available in Project Euclid: 15 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1434377461

Digital Object Identifier
doi:10.1215/00127094-3120167

Mathematical Reviews number (MathSciNet)
MR3357184

Zentralblatt MATH identifier
1334.49149

Subjects
Primary: 47A75: Eigenvalue problems [See also 47J10, 49R05]
Secondary: 49Q20: Variational problems in a geometric measure-theoretic setting 49R05: Variational methods for eigenvalues of operators [See also 47A75] (should also be assigned at least one other classification number in Section 49)

Keywords
Stability for eigenvalues regularity for free boundaries torsional rigidity

Citation

Brasco, Lorenzo; De Philippis, Guido; Velichkov, Bozhidar. Faber–Krahn inequalities in sharp quantitative form. Duke Math. J. 164 (2015), no. 9, 1777--1831. doi:10.1215/00127094-3120167. https://projecteuclid.org/euclid.dmj/1434377461


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