A sharp upper bound for the first Dirichlet eigenvalue and the growth of the isoperimetric constant of convex domains
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- by Pedro Freitas and David Krejčiřík PDF
- Proc. Amer. Math. Soc. 136 (2008), 2997-3006 Request permission
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 Krejčiřík
- Affiliation: Department of Theoretical Physics, Nuclear Physics Institute, Academy of Sciences, 250 68 Řež, Czech Republic
- Email: krejcirik@ujf.cas.cz
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 2997-3006
- MSC (2000): Primary 58J50, 35P15
- DOI: https://doi.org/10.1090/S0002-9939-08-09399-4
- MathSciNet review: 2399068