Chinese Annals of Mathematics, Series B

, Volume 38, Issue 2, pp 497–512 | Cite as

Extremum problems of Laplacian eigenvalues and generalized Polya conjecture



In this survey on extremum problems of Laplacian-Dirichlet eigenvalues of Euclidian domains, the author briefly presents some relevant classical results and recent progress. The main goal is to describe the well-known conjecture due to Polya, its connections to Weyl’s asymptotic formula for eigenvalues and shape optimizations. Many related open problems and some preliminary results are also discussed.


Extremum problems Laplacian eigenvalues Weyl asymptotics Polya’s conjecture Spliting equality Regularity of minimizers 

2000 MR Subject Classification

35 49 57 


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  1. [1]
    Arendt, W., Nittka, R., Peter, W. and Steiner, F., Weyl’s Law, Spectral Properties of the Laplacian in Mathematics and Physics, Math. Anal. of Evolution, Information, and Complexty, Wiley-VCH Verlag GmbH Co. KGaA, Weihreim, 2009, 1–71.CrossRefGoogle Scholar
  2. [2]
    Ashbaush, M. S., The universal eigenvalue bounds of Payne-Polya-Weinberger, Proc. Indian Acad. Sci. Math. Sci., 112, 2002, 3–20.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Bandle, C., Isoperimetric Inequalities and Applications, Monographs and Studies in Math., 7, Pitman, Boston, Mass., London, 1980.Google Scholar
  4. [4]
    Berezin, F., Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR, 37, 1972, 1134–1167 (in Russian); English translation, Math. USSR–Izv., 6, 1972, 1117–1151.MathSciNetGoogle Scholar
  5. [5]
    Birman, M. S. and Solomyak, M. Z., The principle term of spectral asymptotics for “non-smooth” elliptic problems, Functional Anal. Appl., 4, 1970, 1–13.CrossRefGoogle Scholar
  6. [6]
    Bucur, D., Minimization of the k-th eigenvalue of the Dirichlet Laplacian, Arch. Rat. Mech. Anal., 206, 2012, 1073–1083.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    Bucur, D. and Buttazzo, G., Variational methods in shape optimization problems, Progress in Nonlinear Diff. Equ.’s and their Applications, Vol. 65, Birkhäuser, Boston, 2005.Google Scholar
  8. [8]
    Bucur, D. and Dal Maso, G., An existence result of a class of shape optimization problems, Arch. Rat. Mech. Anal., 122, 1993, 183–195.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    Bucur, D. and Henrot, A., Minimization of the third eigenvalue of the Dirichlet Laplacian, R. Soc. London Proc. Ser. A, Math. Phys. Eng. Sci., 456, 2000, 985–996.CrossRefMATHGoogle Scholar
  10. [10]
    Bucur, D., Mazzoleni, D., Pratelli, A. and Velichkov, B., Lipschitz regulairity of the eigenfunctions on optimal domains, Arch. Rat. Mech. Anal., 216, 2015, 117–151.CrossRefMATHGoogle Scholar
  11. [11]
    Caffarelli, L. and Lin, F. H., An optimal partition problem for eigenvalues, J. Sci. Comput., 31, 2007, 5–18.MathSciNetCrossRefGoogle Scholar
  12. [12]
    Caffarelli, L. and Lin, F. H., Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries, J. Amer. Math. Soc., 21, 2008, 847–62.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    Chavel, I., Eigenvalues in Riemannian Geometry, Pure and Appl. Math., 115, Academic Press, New York, 1984.Google Scholar
  14. [14]
    Courant, R. and Hilbert, D., Methods of Mathematical Physics, Vol. 1–2, Wiley, New York, 1953–1962.MATHGoogle Scholar
  15. [15]
    Davies, E. B., Spectral Theory of Differential Operators, Cambridge Univ. Press, Cambridge, 1995.CrossRefMATHGoogle Scholar
  16. [16]
    Guillemin, V. M., Lectures on spectral theory of elliptic operators, Duke Math. J., 44, 1977, 485–517.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    Helffer, B., On spectral minimal partitions: A survey, Milan J. Math., 78, 2010, 575–590.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    Henrot, A., Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Math., Birkhäuser Verlag, Basel-Boston-Berlin, 2006.MATHGoogle Scholar
  19. [19]
    Kac, M., Can one hear the shape of a drum? Amer. Math. Monthly, 73(3), 1966, 1–23.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    Keller, J. B. and Wolf, S. A., Range of the first two eigenvalues of the Laplacian, Proc. R. Soc. London A, 447, 1994, 397–412.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    Kriventsov, D. and Lin, F. H., Regularity for shape optimizer: The non-degenerate case, 2016, preprint.Google Scholar
  22. [22]
    Laptev, A., Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces, J. Funct. Anal., 151, 1997, 531–545.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    Li, P. and Yau, S. T., On the Schrodinger equation and the eigenvalues problems, Comm. Math. Phys., 88, 1983, 309–318.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    Mazzoleni, D. and Pratelli, A., Existence of minimizers for spectral problems, J. Math. Pures Appl., 100, 2013, 433–453.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    Melrose, R., Weyl’s conjecture for manifolds with concave boundary, Proc. Sympos. Pure Math., 36, 1980, 257–274.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    Netrusov, Y. and Safarov, Y., Weyl asymptotic formula for the Laplacian on domains with rough boundaries, Comm. Math. Phys., 253(2), 2005, 481–509.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    Polya, G., On the eigenvalues of vibrating membranes, Proc. London Math. Soc., 11(3), 1961, 419–433.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    Polya, G. and Szegö, G., Isoperimetric Inequalities in Mathematical Physics, AM–27, Princeton Univ. Press, Princeton, 1951.MATHGoogle Scholar
  29. [29]
    Sverak, V., On optimal shape design, J. Math. Pures Appl., 72(6), 1993, 537–551.MathSciNetMATHGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew YorkUSA

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