Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Eigenvalue pinching on convex domains in space forms

Authors: Erwann Aubry, Jérôme Bertrand and Bruno Colbois
Journal: Trans. Amer. Math. Soc. 361 (2009), 1-18
MSC (2000): Primary 35P15, 35P05
Published electronically: August 19, 2008
MathSciNet review: 2439395
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we show that the convex domains of $ \mathbb{H}^n$ which are almost extremal for the Faber-Krahn or the Payne-Polya-Weinberger inequalities are close to geodesic balls. Our proof is also valid in other space forms and allows us to recover known results in $ \mathbb{R}^n$ and $ \mathbb{S}^n$.

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

    A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions.
    Ann. of Math. (2), 135(3):601-628, 1992. MR 1166646 (93d:35105)
    A sharp bound for the ratio of the first two Dirichlet eigenvalues of a domain in a hemisphere of $ S\sp n$.
    Trans. Amer. Math. Soc., 353(3):1055-1087, 2001. MR 1707696 (2001f:35298)
  • 3. A. ÁVILA,
    Stability results for the first eigenvalue of the Laplacian on domains in space forms.
    J. Math. Anal. Appl., 267(2):760-774, 2002. MR 1888036 (2003j:35237)
    A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic space,
    Duke Math. J. 140:245-279, 2007. MR 2359820
  • 5. P. BéRARD,
    Spectral geometry: Direct and inverse problems, volume 1207 of Lecture Notes in Mathematics.
    Springer-Verlag, Berlin, 1986.
    With an appendix by Gérard Besson. MR 861271 (88f:58146)
  • 6. P. BéRARD, D. MEYER, Inégalités isopérimétriques et applications, Ann. Sci. École Norm. Sup. 15(3):513-541, 1982. MR 690651 (84h:58147)
  • 7. I. CHAVEL, Isoperimetric inequalities, Cambridge Tracts in Mathematics, Vol. 145, Cambridge University Press, Cambridge, 2001. MR 1849187 (2002h:58040)
  • 8. K. CHONG, N. RICE,
    Equimeasurable rearrangements of functions.
    Queen's University, Kingston, Ont., 1971.
    Queen's Papers in Pure and Applied Mathematics, n 28. MR 0372140 (51:8357)
  • 9. D. GILBARG, N. TRUDINGER, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Vol. 224, Springer-Verlag, Berlin, 1977. MR 0473443 (57:13109)
  • 10. Q. HAN, F. LIN
    Elliptic partial differential equations. Courant Lecture Notes in Mathematics, 1,
    New York, 1997. MR 1669352 (2001d:35035)
  • 11. L. H¨ORMANDER, Notions of convexity, Progress in Mathematics, Vol. 127, Birkhäuser, Boston, 1994. MR 1301332 (95k:00002)
  • 12. D. JERISON, The first nodal set of a convex domain, Essays in Fourier Analysis in honor of E.M. Stein (C.F. Fefferman, ed.), Princeton Univ. Press, Princeton, NJ, 1993. MR 1315550 (96h:35141)
  • 13. B. KAWOHL,
    Rearrangements and convexity of level sets in PDE, volume 1150 of Lecture Notes in Mathematics.
    Springer-Verlag, Berlin, 1985. MR 810619 (87a:35001)
  • 14. E. LIEB, M. LOSS,
    Analysis. Second edition. Graduate Studies in Mathematics, 14,
    American Mathematical Society, Providence, RI, 2001. MR 1817225 (2001i:00001)
  • 15. P. LI AND S. YAU,
    Estimates of eigenvalues of a compact Riemannian manifold.
    In Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Sympos. Pure Math., XXXVI, pages 205-239. Amer. Math. Soc., Providence, R.I., 1980. MR 573435 (81i:58050)
  • 16. A. MELAS, The stability of some eigenvalue estimates, J. Differential Geom., 36(1):19-33, 1992. MR 1168980 (93d:58178)
  • 17. T. POVEL,
    Confinement of Brownian motion among Poissonian obstacles in $ {\bf R}\sp d, d\ge3$.
    Probab. Theory Related Fields, 114(2):177-205, 1999. MR 1701519 (2000i:60121)
  • 18. M. REED, B. SIMON, Methods of modern mathematical physics IV: Analysis of operators, Academic Press, London (1978). MR 0493421 (58:12429c)
  • 19. M. SHUBIN, Spectral theory of the Schrödinger operators on non-compact manifolds: Qualitative results, In Spectral Theory and Geometry (Edinburgh, 1998), Vol. 273 of London Math. Soc. Lecture Note Ser., pages 226-283. Cambridge Univ. Press, Cambridge, (1999). MR 1736869 (2001d:58037)
  • 20. M. STRUWE, Variational Methods, Ergeb. Math. Grenz., Vol. 34, Springer-Verlag, Berlin (2000). MR 1736116 (2000i:49001)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35P15, 35P05

Retrieve articles in all journals with MSC (2000): 35P15, 35P05

Additional Information

Erwann Aubry
Affiliation: Laboratoire J.-A. Dieudonné, Université de Nice Sophia-Antipolis, UMR6621 (UNSA-CNRS), Parc Valrose, F-06108 Nice Cedex, France

Jérôme Bertrand
Affiliation: Institut de Mathématiques, Université de Toulouse of CNRS, UMR 5219, 118, route de Narbonne, F-31062 Toulouse, Cedex 4, France

Bruno Colbois
Affiliation: Institut de mathématiques, Université de Neuchâtel, Rue Émile Argand, 11, Case postale 158, CH-2009 Neuchâtel, Switzerland

Received by editor(s): April 26, 2006
Published electronically: August 19, 2008
Additional Notes: The first author was partially supported by FNRS Swiss Grant N. 20-101469.
Article copyright: © Copyright 2008 American Mathematical Society

American Mathematical Society