Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format 		Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

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

Author(s): Pedro Freitas; David Krejcirík
Journal: Proc. Amer. Math. Soc. 136 (2008), 2997-3006.
MSC (2000): Primary 58J50, 35P15
Posted: April 7, 2008
MathSciNet review: 2399068
Retrieve article in: PDF

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 $ 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.

References:

[A]
M. I. Aissen, A set function defined for convex plane domaines, Pacific J. Math. 8 (1958), 383-399. MR 0123968 (23:A1289)

[AF1]
P. Antunes and P. Freitas, New bounds for the principal Dirichlet eigenvalue of planar regions, Exp. Math. 15 (2006), 333-342. MR 2264470 (2007e:35039)

[AF2]
P. Antunes and P. Freitas, A numerical study of the spectral gap, J. Phys. A: Math. Theor. 41 (2008), 055201.

[AM1]
T. M. Apostol and M. A. Mnatsakanian, Polygons circumscribing circles, Amer. Math. Month. 111 (2004), 853-863. MR 2104691 (2005g:51025)

[AM2]
T. M. Apostol and M. A. Mnatsakanian, Solids circumscribing spheres, Amer. Math. Month. 113 (2006), 521-540. MR 2231137 (2007c:51018)

[AB]
M. Ashbaugh and R. Benguria, A sharp bound for the ratio of the first two eigenvalues of the Dirichelt Laplacian and extensions, Ann. Math. 135 (1992), 601-628. MR 1166646 (93d:35105)

[BC]
J. Bertrand and B. Colbois, Capacité et inéqualité de Faber-Krahn dans $ \mathbb{R}^{n}$, J. Funct. Anal. 232 (2006), 1-28. MR 2200165 (2007c:31005)

[BZ]
Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Springer, Berlin, 1988. MR 936419 (89b:52020)

[EE]
D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Oxford University Press, New York, 1987. MR 929030 (89b:47001)

[FMP]
N. Fusco, F. Maggi and A. Pratelli, Stability estimates for certain Faber-Krahn, iso- capacitary and Cheeger inequalities, to appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze.

[G]
H. Guggenheimer, Concave solutions of a Dirichlet problem, Proc. Amer. Math. Soc. 40 (1973), 501-506. MR 0330481 (48:8818)

[HS]
J. Hersch and W. Sawyer, Numerical implementation of coherence for the example of the ``Swiss cross'', Numer. Math. 59 (1991), 659-665. MR 1128461 (92j:65162)

[MS]
V. Maz'ya and M. Shubin, Can one see the fundamental frequency of a drum?, Lett. Math. Phys. 74 (2005), 135-151. MR 2191951 (2006m:58050)

[M]
A. Melas, The stability of some eigenvalue estimates, J. Differential Geom. 36 (1992), 19-33. MR 1168980 (93d:58178)

[PW]
L. E. Payne and H. F. Weinberger, Some isoperimetric inequalities for membrane frequencies and torsional rigidity, J. Math. Anal. Appl. 2 (1961), 210-216. MR 0149735 (26:7220)

[P]
G. Pólya, Two more inequalities between physical and geometrical quantities, J. Indian Math. Soc. (N.S.) 24 (1960), 413-419 (1961). MR 0133059 (24:A2895)

[PS]
G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, 27, Princeton University Press, Princeton, NJ, 1951. MR 0043486 (13:270d)

[Po]
A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC, Boca Raton, FL, 2002. MR 1935578 (2003i:35001)

[Pr]
M. H. Protter, A lower bound for the fundamental frequency of a convex region, Proc. Amer. Math. Soc. 81 (1981), 65-70. MR 589137 (82b:35113)

[S]
A. Savo, Lower bounds for the nodal length of eigenfunctions of the Laplacian, Ann. Global Anal. Geom. 19 (2001), 133-151. MR 1826398 (2002g:58055)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 58J50, 35P15

Retrieve articles in all Journals with MSC (2000): 58J50, 35P15

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 Rez, Czech Republic
Email: krejcirik@ujf.cas.cz

DOI: 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
Posted: 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 of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia