SIAM Journal on Mathematical Analysis


SIAM J. Math. Anal., 39(4), 1191–1207. (17 pages)

Uniqueness in the Faber–Krahn Inequality for Robin Problems

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Article Data

History

Submitted: 21 November 2006
Accepted: 30 March 2007
Published online: 21 November 2007

AMS Subject Headings

35P15, 35J25

Publication Data

ISSN (print): 0036-1410
ISSN (online): 1095-7154
Publisher: Society for Industrial and Applied Mathematics
CODEN: sjmaah
Daniel Daners and James Kennedy
https://doi.org/10.1137/060675629

We prove uniqueness in the Faber–Krahn inequality for the first eigenvalue of the Laplacian with Robin boundary conditions, asserting that among all sufficiently smooth domains of fixed volume, the ball is the unique minimizer for the first eigenvalue. The method of proof, which avoids the use of any symmetrization, also works in the case of Dirichlet boundary conditions. We also give a characterization of all symmetric elliptic operators in divergence form whose first eigenvalue is minimal.

Copyright © 2007 Society for Industrial and Applied Mathematics
Permalink: https://doi.org/10.1137/060675629

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