Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

An isoperimetric inequality for the second eigenvalue of the Laplacian with Robin boundary conditions


Author: James Kennedy
Journal: Proc. Amer. Math. Soc. 137 (2009), 627-633
MSC (2000): Primary 35P15, 35J25
DOI: https://doi.org/10.1090/S0002-9939-08-09704-9
Published electronically: October 8, 2008
MathSciNet review: 2448584
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the second eigenvalue of the Laplacian with Robin boundary conditions is minimized among all bounded Lipschitz domains of fixed volume by the domain consisting of the disjoint union of two balls of equal volume.


References [Enhancements On Off] (What's this?)

  • 1. M.-H. Bossel, Membranes élastiquement liées: Extension du théorème de Rayleigh-Faber-Krahn et de l'inégalité de Cheeger, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), 47-50. MR 0827106 (87f:35186)
  • 2. H. Brezis, Analyse fonctionnelle. Théorie et applications, Masson, Paris, 1983. MR 0697382 (85a:46001)
  • 3. R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. 1, Interscience Publishers, New York, 1953. MR 0065391 (16:426a)
  • 4. E. N. Dancer and D. Daners, Domain perturbation for elliptic equations subject to Robin boundary conditions, J. Differential Equations 138 (1997), 86-132. MR 1458457 (98e:35017)
  • 5. D. Daners, A Faber-Krahn inequality for Robin problems in any space dimension, Math. Ann. 335 (2006), 767-785. MR 2232016 (2007b:35242)
  • 6. D. Daners, Robin boundary value problems on arbitrary domains, Trans. Amer. Math. Soc. 352 (2000), 4207-4236. MR 1650081 (2000m:35048)
  • 7. D. Daners and J. Kennedy, Uniqueness in the Faber-Krahn inequality for Robin problems, SIAM J. Math. Anal. 39 (2007/08), 1191-1207. MR 2368899
  • 8. D. E. Edmunds and W. D. Evans, Spectral theory and differential operators. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1987. MR 0929030 (89b:47001)
  • 9. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften, vol. 224, Springer-Verlag, Berlin, 1983. MR 0737190 (86c:35035)
  • 10. A. Henrot, Minimization problems for eigenvalues of the Laplacian, J. Evol. Equ. 3 (2003), 443-461. MR 2019029 (2005a:49078)
  • 11. M. W. Hirsch, Differential topology. Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362 (56:6669)
  • 12. A. A. Lacey, J. R. Ockendon and J. Sabina, Multidimensional reaction diffusion equations with nonlinear boundary conditions, SIAM J. Appl. Math. 58 (1998), 1622-1647. MR 1637882 (99g:35066)
  • 13. Shape Analysis for Eigenvalues. Abstracts from the workshop held April 8-14, 2007. Organised by D. Bucur, G. Buttazzo and A. Henrot, Oberwolfach Reports, vol. 4, no. 2 (2007), 995-1026.
  • 14. M. Warma, The Robin and Wentzell-Robin Laplacians on Lipschitz domains, Semigroup Forum 73 (2006), 10-30. MR 2277314 (2007g:35032)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35P15, 35J25

Retrieve articles in all journals with MSC (2000): 35P15, 35J25


Additional Information

James Kennedy
Affiliation: School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia
Email: J.Kennedy@maths.usyd.edu.au

DOI: https://doi.org/10.1090/S0002-9939-08-09704-9
Keywords: Isoperimetric inequality, Laplacian, Robin boundary conditions, elastically supported membrane
Received by editor(s): January 30, 2008
Published electronically: October 8, 2008
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society