Advertisement

Mathematische Zeitschrift

, Volume 278, Issue 1–2, pp 529–546 | Cite as

Extremal eigenvalues of the Laplacian on Euclidean domains and closed surfaces

  • Bruno Colbois
  • Ahmad El SoufiEmail author
Article

Abstract

We investigate properties of the sequences of extremal values that could be achieved by the eigenvalues of the Laplacian on Euclidean domains of unit volume, under Dirichlet and Neumann boundary conditions, respectively. In a second part, we study sequences of extremal eigenvalues of the Laplace–Beltrami operator on closed surfaces of unit area.

Keywords

Laplacian eigenvalues Extremal eigenvalues Spectral geometry  Closed surfaces 

Mathematics Subject Classification (1991)

35P15 58J50 58E11 

Notes

Acknowledgments

The authors would like to thank the referee for his valuable comments.

References

  1. 1.
    Anné, C.: Spectre du laplacien et écrasement d’anses. Ann. Sci. École Norm. Sup. (4) 20(2), 271–280 (1987)zbMATHGoogle Scholar
  2. 2.
    Antunes, P.R.S., Freitas, P.: Numerical optimization of low eigenvalues of the Dirichlet and Neumann Laplacians. J. Optim. Theory Appl. 154(1), 235–257 (2012)CrossRefzbMATHGoogle Scholar
  3. 3.
    Berezin, F.A.: Covariant and contravariant symbols of operators. Izv. Akad. Nauk SSSR Ser. Mat. 36, 1134–1167 (1972)zbMATHGoogle Scholar
  4. 4.
    Bucur, D.: Minimization of the \(k\)-th eigenvalue of the Dirichlet Laplacian. Arch. Ration. Mech. Anal. 206(3), 1073–1083 (2012)CrossRefzbMATHGoogle Scholar
  5. 5.
    Colbois, B., Dryden, E.B., El Soufi, A.: Bounding the eigenvalues of the Laplace-Beltrami operator on compact submanifolds. Bull. Lond. Math. Soc. 42(1), 96–108 (2010)CrossRefzbMATHGoogle Scholar
  6. 6.
    Colbois, B., El Soufi, A.: Extremal eigenvalues of the Laplacian in a conformal class of metrics: the ‘conformal spectrum’. Ann. Glob. Anal. Geom. 24(4), 337–349 (2003)CrossRefzbMATHGoogle Scholar
  7. 7.
    Dodziuk, J.: Eigenvalues of the Laplacian on forms. Proc. Am. Math. Soc. 85(3), 437–443 (1982)CrossRefzbMATHGoogle Scholar
  8. 8.
    El Soufi, A., Ilias, S.: Le volume conforme et ses applications d’après Li et Yau. In: Séminaire de Théorie Spectrale et Géométrie, Année 1983–1984, pp. VII.1–VII.15. Univ. Grenoble I, Institut Fourier (1984)Google Scholar
  9. 9.
    El Soufi, A., Giacomini, H., Jazar, M.: A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle. Duke Math. J. 135(1), 181–202 (2006)CrossRefzbMATHGoogle Scholar
  10. 10.
    El Soufi, A., Ilias, S.: Riemannian manifolds admitting isometric immersions by their first eigenfunctions. Pac. J. Math. 195(1), 91–99 (2000)CrossRefzbMATHGoogle Scholar
  11. 11.
    El Soufi, A., Ilias, S.: Extremal metrics for the first eigenvalue of the Laplacian in a conformal class. Proc. Am. Math. Soc. 131(5), 1611–1618 (2003). (electronic)CrossRefzbMATHGoogle Scholar
  12. 12.
    Girouard, A., Nadirashvili, N., Polterovich, I.: Maximization of the second positive Neumann eigenvalue for planar domains. J. Differ. Geom. 83(3), 637–661 (2009)zbMATHGoogle Scholar
  13. 13.
    Hassannezhad, A.: Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem. J. Funct. Anal. 261(12), 3419–3436 (2011)CrossRefzbMATHGoogle Scholar
  14. 14.
    Henrot, A.: Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006)Google Scholar
  15. 15.
    Hersch, J.: Quatre propriétés isopérimétriques de membranes sphériques homogènes. C. R. Acad. Sci. Paris Sér. A-B 270, A1645–A1648 (1970)Google Scholar
  16. 16.
    Jakobson, D., Levitin, M., Nadirashvili, N., Nigam, N., Polterovich, I.: How large can the first eigenvalue be on a surface of genus two? Int. Math. Res. Not. 63, 3967–3985 (2005)CrossRefGoogle Scholar
  17. 17.
    Jakobson, D., Nadirashvili, N., Polterovich, I.: Extremal metric for the first eigenvalue on a Klein bottle. Can. J. Math. 58(2), 381–400 (2006)CrossRefzbMATHGoogle Scholar
  18. 18.
    Karpukhin, M.: Maximization of the first nontrivial eigenvalue on the surface of genus two. arXiv:1309.5057[math.DG], pp. 1–9 (2013)
  19. 19.
    Korevaar, N.: Upper bounds for eigenvalues of conformal metrics. J. Differ. Geom. 37(1), 73–93 (1993)zbMATHGoogle Scholar
  20. 20.
    Kröger, P.: Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space. J. Funct. Anal. 106(2), 353–357 (1992)CrossRefzbMATHGoogle Scholar
  21. 21.
    Kröger, P.: Estimates for sums of eigenvalues of the Laplacian. J. Funct. Anal. 126(1), 217–227 (1994)CrossRefzbMATHGoogle Scholar
  22. 22.
    Kuiper, N.H.: On \(C^1\)-isometric imbeddings. I, II. Nederl. Akad. Wetensch. Proc. Ser. A. 58 Indag. Math. 17(545–556), 683–689 (1955)Google Scholar
  23. 23.
    Li, P., Yau, S.T.: A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. Invent. Math. 69(2), 269–291 (1982)CrossRefzbMATHGoogle Scholar
  24. 24.
    Li, P., Yau, S.T.: On the Schrödinger equation and the eigenvalue problem. Commun. Math. Phys. 88(3), 309–318 (1983)CrossRefzbMATHGoogle Scholar
  25. 25.
    Mazzoleni, D., Pratelli, A.: Existence of minimizers for spectral problems. J. Math. Pures Appl. (9) 100(3), 433–453 (2013)CrossRefzbMATHGoogle Scholar
  26. 26.
    Nadirashvili, N.: Berger’s isoperimetric problem and minimal immersions of surfaces. Geom. Funct. Anal. 6(5), 877–897 (1996)CrossRefzbMATHGoogle Scholar
  27. 27.
    Nadirashvili, N.: Isoperimetric inequality for the second eigenvalue of a sphere. J. Differ. Geom. 61(2), 335–340 (2002)zbMATHGoogle Scholar
  28. 28.
    Oudet, É.: Numerical minimization of eigenmodes of a membrane with respect to the domain. ESAIM Control Optim. Calc. Var. 10(3), 315–330 (2004). (electronic)CrossRefzbMATHGoogle Scholar
  29. 29.
    Poliquin, G., Roy-Fortin, G.: Wolf-Keller theorem for Neumann eigenvalues. Ann. Sci. Math. Québec 36(1):169–178 (2013), 2012.Google Scholar
  30. 30.
    van den Berg, M., Iversen, M.: On the minimization of Dirichlet eigenvalues of the Laplace operator. J. Geom. Anal. 23(2), 660–676 (2013)CrossRefzbMATHGoogle Scholar
  31. 31.
    Wolf, Sven Andreas, Keller, Joseph B.: Range of the first two eigenvalues of the Laplacian. Proc. Roy. Soc. Lond. Ser. A 447(1930), 397–412 (1994)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Laboratoire de MathématiquesUniversité de NeuchâtelNeuchâtelSwitzerland
  2. 2.Laboratoire de Mathématiques et Physique Théorique, UMR-CNRS 7350Université de ToursToursFrance

Personalised recommendations