Sharp Stability of Some Spectral Inequalities

In this work we review two classical isoperimetric inequalities involving eigenvalues of the Laplacian, both with Dirichlet and Neumann boundary conditions. The first one is classically attributed to Krahn and P. Szego and asserts that among sets of given measure, the disjoint union of two balls wit...

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Bibliographic Details
Published in:	Geometric and functional analysis Vol. 22; no. 1; pp. 107 - 135
Main Authors:	Brasco, Lorenzo, Pratelli, Aldo
Format:	Journal Article
Language:	English
Published:	Basel SP Birkhäuser Verlag Basel 01-02-2012
Subjects:	Analysis Mathematics Mathematics and Statistics Stability for eigenvalues Krahn–Szego inequality 49R05 47A75 Szegő–Weinberger inequality isoperimetric inequalities 49Q20
Online Access:	Get full text
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Summary:	In this work we review two classical isoperimetric inequalities involving eigenvalues of the Laplacian, both with Dirichlet and Neumann boundary conditions. The first one is classically attributed to Krahn and P. Szego and asserts that among sets of given measure, the disjoint union of two balls with the same radius minimizes the second eigenvalue of the Dirichlet–Laplacian, while the second one is due to G. Szegő and Weinberger and deals with the maximization of the first non-trivial eigenvalue of the Neumann–Laplacian. New stability estimates are provided for both of them.
ISSN:	1016-443X 1420-8970
DOI:	10.1007/s00039-012-0148-9