A sharp upper bound for the first Dirichlet eigenvalue and the growth of the isoperimetric constant of convex domains

Authors:
Pedro Freitas and David Krejcirík

Journal:
Proc. Amer. Math. Soc. **136** (2008), 2997-3006

MSC (2000):
Primary 58J50, 35P15

Published electronically:
April 7, 2008

MathSciNet review:
2399068

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Abstract | References | Similar Articles | Additional Information

Abstract: We show that as the ratio between the first Dirichlet eigenvalues of a convex domain and of the ball with the same volume becomes large, the same must happen to the corresponding ratio of isoperimetric constants. The proof is based on the generalization to arbitrary dimensions of Pólya and Szegö's upper bound for the first eigenvalue of the Dirichlet Laplacian on planar star-shaped domains which depends on the support function of the domain.

As a by-product, we also obtain a sharp upper bound for the spectral gap of convex domains.

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Additional Information

**Pedro Freitas**

Affiliation:
Department of Mathematics, Faculdade de Motricidade Humana (TU Lisbon) and Group of Mathematical Physics, University of Lisbon, Complexo Interdisciplinar, Av. Prof. Gama Pinto 2, P-1649-003 Lisboa, Portugal

Email:
freitas@cii.fc.ul.pt

**David Krejcirík**

Affiliation:
Department of Theoretical Physics, Nuclear Physics Institute, Academy of Sciences, 25068 Řež, Czech Republic

Email:
krejcirik@ujf.cas.cz

DOI:
http://dx.doi.org/10.1090/S0002-9939-08-09399-4

Received by editor(s):
March 20, 2007

Received by editor(s) in revised form:
January 24, 2008

Published electronically:
April 7, 2008

Additional Notes:
This work was partially supported by FCT, Portugal, through programs POCTI/MAT/60863/2004, POCTI/POCI2010 and SFRH/BPD/11457/2002. The second author was also supported by the Czech Academy of Sciences and its Grant Agency within the projects IRP AV0Z10480505 and A100480501, and by the project LC06002 of the Ministry of Education, Youth and Sports of the Czech Republic.

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.