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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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: 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 $ 1951$ 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
PII: S 0002-9939(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.