Skip to main content
SpringerLink
Log in
Book cover

Nonlinear Evolution Equations and Related Topics pp 443–461Cite as

Birkhäuser

Minimization problems for eigenvalues of the Laplacian

  • Chapter

Abstract

This paper is a survey on classical results and open questions about minimization problems concerning the lower eigenvalues of the Laplace operator. After recalling classical isoperimetric inequalities for the two first eigenvalues, we present recent advances on this topic. In particular, we study the minimization of the second eigenvalue among plane convex domains. We also discuss the minimization of the third eigenvalue. We prove existence of a minimizer. For others eigenvalues, we just give some conjectures. We also consider the case of Neumann, Robin and Stekloff boundary conditions together with various functions of the eigenvalues.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   109.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alessandrini, G.Nodal lines of eigenfunctions of the fixed membrane problem in general convex domainsComment. Math. HeIv.69(1994) no. 1, 142–154.

    Article  MathSciNet  MATH  Google Scholar 

  2. Ashbaugh, M. S.Open problems on eigenvalues of the LaplacianAnalytic and Geometric Inequalities and Their Applications, T. M. Rassias and H. M. Srivastava (editors), vol. 4787, Kluwer 1999.

    Google Scholar 

  3. Ashbaugh, M. S. and Benguria, R.Proof of the Payne-Pólya-Weinberger conjectureBull. Amer. Math. Soc.25(1991) n°1,19–29.

    Article  MathSciNet  MATH  Google Scholar 

  4. Ashbaugh, M. S. and Benguria, R., Asharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensionsAnn. of Math.135(1992) no. 3, 601–628.

    Article  MathSciNet  MATH  Google Scholar 

  5. ASHBAUGH, M. S. and Benguria, R.On Rayleigh’s conjecture for the clamped plate and its generalization to three dimensionsDuke Math. J.78(1995), 1–17.

    Article  MathSciNet  MATH  Google Scholar 

  6. Bandle, C.Isoperimetric inequalities and applications.Monographs and Studies in Mathematics, 7. Pitman, Boston, Mass.-London 1980.

    Google Scholar 

  7. Bossel, M. H.Membranes élastiquement liées: extension du théorème de Rayleigh-Faber-Krahn et de l’inégalité de CheegerC. R. Acad. Sci. Paris Sér. I Math.302(1986) no. 1,47–50.

    MathSciNet  MATH  Google Scholar 

  8. Brock, E, Anisoperimetric inequality for eigenvalues of the Stekloff problemZ. Angew. Math. Mech.81(2001) no. I, 69–71.

    Article  MathSciNet  Google Scholar 

  9. Bucur, D., BUTTAZZO, G. and Ftgue!redo, I.On the attainable eigenvalues of the Laplace operatorSIAM J. Math. Anal.30(1999) no. 3,527–536.

    Article  MathSciNet  MATH  Google Scholar 

  10. Bucur, D. and Henrot, A.Minimization of the third eigenvalue of the Dirichlet LaplacianProc. Roy. Soc. London456(2000), 985–996.

    Article  MathSciNet  MATH  Google Scholar 

  11. Bucur, D. and Zolesio, J. P.N-dimensional shape optimization under capacitary constraintsJ. of Diff. Eq.123(1995), n°2,504–522.

    Article  MathSciNet  MATH  Google Scholar 

  12. BUTTAZZO, G. and Dal Maso, G., An Existence Result for a Class of Shape Optimization ProblemsArch. Rational Mech. Anal.122(1993), 183–195.

    Article  MathSciNet  MATH  Google Scholar 

  13. Chatelain, T. and Choullt, M., Clarke generalized gradient for eigenvaluesCommun. Appl_ Anal. 1 (1997) no. 4,443–454.

    MATH  Google Scholar 

  14. Cox, S. J.The generalized gradient at a multiple eigenvalueJ. Funct. Anal.133(1995) no. 1, 30–40.

    Article  MathSciNet  MATH  Google Scholar 

  15. Cox, S. J. and Ross, M.Extremal eigenvalue problems for starlike planar domainsJ. Differential Equations120 (1995)174–197.

    Google Scholar 

  16. COXS.J. and Ross, M.The maximization of Neumann eigenvalues on convex domainsto appear.

    Google Scholar 

  17. Courant, R. and Hilbert, D.Methods of Mathematical Physicsvol. 1 et 2, Wiley, New York 1953 et 1962.

    Google Scholar 

  18. Dal Maso, G.An introduction to I’ -convergenceBirkhäuser, Boston 1993.

    Google Scholar 

  19. Daners, D.Robin boundary value problems on arbitrary domainsTrans. AMS352(2000), 4207–4236.

    Article  MathSciNet  MATH  Google Scholar 

  20. Dautray, R. and Lions, J. L., (ed)Analyse mathématique et calcul numérique Vol.I and II, Masson, Paris 1984.

    Google Scholar 

  21. Faber, G.Beweisdass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibtSitz. Ber. Bayer. Akad. Wits. 1923, 169–172.

    Google Scholar 

  22. Flucher, M., Approximation of Dirichlet eigenvalues on domains with small holesJ. Math. Anal. Appl.193(1995) no. 1, 169–199.

    Article  MathSciNet  MATH  Google Scholar 

  23. HARRELL, E. M., KROGER, P. and KURATA, K.On the placement of an obstacle or a well so as to optimize the fundamental eigenvalueto appear in SIAM J. Math. Anal.

    Google Scholar 

  24. Henrot, A. and Oudet, E.Le stade ne minimise pas)L2parmi les ouverts convexes du planC. R. Acad. Sci. Paris Sr. I Math, 332 (2001) no. 4,275–280.

    MathSciNet  Google Scholar 

  25. Henrot, A. and OUDFT, E.Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditionsto appear.

    Google Scholar 

  26. Henrot, A. and Pierre, M.Optimisation de formebook to appear.

    Google Scholar 

  27. Hersch, J.The method of interior parallels applied to polygonal or multiply connected membranesPacific J. Math.13(1963), 1229–1238.

    Article  MathSciNet  MATH  Google Scholar 

  28. Hersch, J.Contraintes rectilignes parallels et valeurs propres de membranes vibrantesZ. Angew. Math. Phys.17 (1966)457–460.

    Article  MathSciNet  MATH  Google Scholar 

  29. Hersch, J. and Payne, L. E.Extremal principles and isoperimetric inequalities for some mixed problems of Stekloff’s typeZ. Angew. Math. Phys. 19 (1968), 802–817.

    Article  MathSciNet  MATH  Google Scholar 

  30. Hersch, J., Payne, L. E. and Schinier, M. M.Some inequalities for Stekloff eigenvaluesArch. Rational Mech. Anal. 57 (1975), 99–114.

    MathSciNet  Google Scholar 

  31. Kawohl, B., Rearrangements and convexity of level sets in Pde, Lecture Notes in Mathematics, 1150. Springer-Verlag, Berlin, 1985.

    Google Scholar 

  32. Kawohl, B., Pironneau, O., Tartar, L. and Zolsio, J. P., Optimal shape design, Lecture Notes in Mathematics, 1740. (Lectures given at the Joint C.I.M./C.I.M.E. Summer School held in Tróia, June 1–6, 1998, Edited by A. Cellina and A. Ornelas).

    Google Scholar 

  33. KESAVAN, S.On two functionals connected to the Laplacian in a class of doubly connected domainsto appear.

    Google Scholar 

  34. Krahn, E.Ober eine von Rayleigh formulierte Minimaleigenschaft des KreisesMath. Ann.94(1924), 97–100.

    Article  MathSciNet  Google Scholar 

  35. Krahn, E.Ober Minimaleigenschaften der Kugel in drei un mehr DimensionenActa Comm. Univ. Dorpat. A9 (1926), 1–44.

    Google Scholar 

  36. Melas, A.On the nodal line of the second eigenfunction of the Laplacian in R 2J. Diff. Geometry.35(1992), 255–263.

    MathSciNet  MATH  Google Scholar 

  37. Nadirashvill N. S.Rayleigh’s conjecture on the principal frequency of the clamped plateArch. Rational Mech. Anal.129(1995), 1–10.

    Article  MathSciNet  Google Scholar 

  38. Osserman, R.The isoperimetric inequalityBull. AMS84(1978) no. 6,1182–1238.

    Article  MathSciNet  MATH  Google Scholar 

  39. Osserman, R.Isoperimetric inequalities and eigenvalues of the LaplacianProceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 435–442, Acad. Sci. Fennica, Helsinki, 1980.

    Google Scholar 

  40. Oudet, E.Some numerical results about minimization problems involving eigenvaluesto appear.

    Google Scholar 

  41. Payne, L. E.Isoperimetric inequalities and their applicationsSIAM Rev.9(1967), 453–488.

    Article  MathSciNet  MATH  Google Scholar 

  42. Payne, L. E.Some comments on the past fifty years of isoperimetric inequalitiesInequalities (Birmingham, 1987), 143–161, Lecture Notes in Pure and Appl. Math.129Dekker, New York 1991.

    Google Scholar 

  43. Payne, L. E., PÓlya, G. and Weinberger, H. F.On the ratio of consecutive eigenvaluesJ. Math. Phys.35(1956), 289–298.

    MATH  Google Scholar 

  44. Payne, L. E., Payne, L. E. and Schaefer, P. W.Eigenvalue and eigenfunction inequalities for the elastically supported membraneZ. Angew. Math. Phys.52(2001) no. 5, 888–895.

    Article  MathSciNet  MATH  Google Scholar 

  45. Payne, L. E. and Weinberger, H. F.An optimal Poincaré inequality for convex domainsArch. Rational Mech. Anal.5(1960), 286–292.

    Article  MathSciNet  MATH  Google Scholar 

  46. Payne, L. E. and Weinberger, H. F.Some isoperimetric inequalities for membrane frequencies and torsional rigidityJ. Math, Anal. Appl.2(1961), 210–216.

    Article  MathSciNet  MATH  Google Scholar 

  47. PÓlya, G.On the characteristic frequencies of a symmetric membraneMath. Z. 63 (1955), 331–337.

    Article  MathSciNet  MATH  Google Scholar 

  48. Polya, G. and SzegÖ, G.Isoperimetric inequalities in mathematical physicsAnn. Math. Studies27Princeton Univ. Press, 1951.

    MATH  Google Scholar 

  49. Rassias, T.The isoperimetric inequality and eigenvalues of the Laplacian. Constantin Carathodory: an international tributevol. I, II, 1146–1163, World Sci. Publishing, Teaneck, NJ, 1991.

    Google Scholar 

  50. Schoen, R. and Yau, S. -T.Lectures on differential geometryConference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge, MA, 1994.

    Google Scholar 

  51. Serrin, J.A symmetry problem in potential theoryArch. Rational Mech. Anal.43(1971), 304–318.

    Article  MathSciNet  MATH  Google Scholar 

  52. Simon, J.Differentiation with respect to the domain in boundary value problemsNum. Funct. Anal. Optimz. 2 (1980), 649–687.

    Article  MATH  Google Scholar 

  53. Sokolowski, J. and Zolesio, J. P.Introduction to shape optimization: shape sensitity analysisSpringer Series in Computational Mathematics, Vol. 10, Springer, Berlin 1992.

    Google Scholar 

  54. SzegÖ, G.Inequalities for certain eigenvalues of a membrane of given areaJ. Rational Mech. Anal. 3 (1954), 343–356.

    MathSciNet  MATH  Google Scholar 

  55. Troesch, B. A.Elliptical Membranes with smallest second eigenvalueMath. of Computation, 27–124 (1973), 767–772.

    Google Scholar 

  56. Weinberger, H. F.An isoperimetric inequality for the N -dimensionalfree membrane problemJ. Rational Mech. Anal.5(1956), 633–636.

    MathSciNet  MATH  Google Scholar 

  57. Weinstock, R.Inequalities for a classical eigenvalue problemJ. Rational Mech. Anal. 3 (1954), 745–753.

    MathSciNet  MATH  Google Scholar 

  58. Wolf, S. A. and Keller, J. B.Range of the first two eigenvalues of the LaplacianProc. R. Soc. London A447(1994), 397–412.

    Article  MathSciNet  MATH  Google Scholar 

  59. Yau, S. -T.Problem sectionSeminar on Differential Geometry, pp. 669–706, Ann. of Math. Stud. 102, Princeton Univ. Press, Princeton, N.J. 1982.

    Google Scholar 

  60. Yau, S. -T.Open problems in geometry. Differential geometry: partial differential equations on manifolds(Los Angeles, CA, 1990), 1–28, Proc. Sympos. Pure Math. 54, Part 1, Amer. Math. Soc. Providence, RI 1993.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ecole des Mines and Institut Elie Cartan Nancy UMR 7502 CNRS and projet Corida, INRIA, B.P. 239, Vandœuvre-lès-Nancy, 54506, France

    Antoine Henrot

Authors
  1. Antoine Henrot
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Applied Analysis, University of Ulm, 89069, Ulm, Germany

    Wolfgang Arendt

  2. Analyse Numérique, Université Pierre et Marie Curie, B.C. 187, 4, place Jussieu, 75252, Paris Cedex 05, France

    Haïm Brézis

  3. Department of Mathematics Hill Center, Busch Campus, Rutgers University, 110 Frelinghuysen RD, Piscataway, NJ, 08854, USA

    Haïm Brézis

  4. Campus de Ker Lann, Antenne de Bretagne de l’ENS Cachan, 35170, Bruz, France

    Michel Pierre

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer Basel AG

About this chapter

Cite this chapter

Henrot, A. (2003). Minimization problems for eigenvalues of the Laplacian. In: Arendt, W., Brézis, H., Pierre, M. (eds) Nonlinear Evolution Equations and Related Topics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7924-8_24

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-7924-8_24

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-7643-7107-4

  • Online ISBN: 978-3-0348-7924-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation