Skip to main content

Analyticity and Criticality Results for the Eigenvalues of the Biharmonic Operator

  • Conference paper
  • First Online:
Geometric Properties for Parabolic and Elliptic PDE's (GPPEPDEs 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 176))

Abstract

We consider the eigenvalues of the biharmonic operator subject to several homogeneous boundary conditions (Dirichlet, Neumann, Navier, Steklov). We show that simple eigenvalues and elementary symmetric functions of multiple eigenvalues are real analytic, and provide Hadamard-type formulas for the corresponding shape derivatives. After recalling the known results in shape optimization, we prove that balls are always critical domains under volume constraint.

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

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover 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

Institutional subscriptions

Similar content being viewed by others

References

  1. Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Commun. Pure Appl. Math. 12, 623–727 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  2. Antunes, P.R.S., Gazzola, F.: Convex shape optimization for the least biharmonic Steklov eigenvalue. ESAIM, Control Optim. Calc. Var. 19(2), 385–403 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ashbaugh, M.S., Benguria, R.D.: On Rayleigh’s conjecture for the clamped plate and its generalization to three dimensions. Duke Math. J. 78(1), 1–7 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. Babuška, I.: Stabilität des Definitionsgebietes mit Rücksicht auf grundlegende Probleme der Theorie der partiellen Differentialgleichungen auch im Zusammenhang mit der Elastizitätstheorie. I, II (Russian, German summary). Czechoslovak Math. J. 11(86), 76–105, 165–203 (1961)

    Google Scholar 

  5. Berchio, E., Gazzola, F., Mitidieri, E.: Positivity preserving property for a class of biharmonic elliptic problems. J. Differ. Equ. 229(1), 1–23 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bucur, D., Ferrero, A., Gazzola, F.: On the first eigenvalue of a fourth order Steklov problem. Calc. Var. Partial Differ. Equ. 35(1), 103–131 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bucur, D., Gazzola, F.: The first biharmonic Steklov eigenvalue: positivity preserving and shape optimization. Milan J. Math. 79(1), 247–258 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Buoso, D.: Shape sensitivity analysis of the eigenvalues of polyharmonic operators and elliptic systems. Ph.D. Thesis, Università degli Studi di Padova, Padova (2015)

    Google Scholar 

  9. Buoso, D.: Shape differentiability of the eigenvalues of elliptic systems. In: Integral Methods in Science and Engineering: Theoretical and Computational Advances. Birkhäuser, Basel (2015)

    Google Scholar 

  10. Buoso, D., Chasman, L.M., Provenzano, L.: On the stability of some isoperimetric inequalities for the fundamental tones of free plates. J. Spectral Theor. submitted

    Google Scholar 

  11. Buoso, D., Lamberti, P.D.: Eigenvalues of polyharmonic operators on variable domains. ESAIM Control Optim. Calc. Var. 19(4), 1225–1235 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Buoso, D., Lamberti, P.D.: Shape deformation for vibrating hinged plates. Math. Methods Appl. Sci. 37(2), 237–244 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Buoso, D., Lamberti, P.D.: Shape sensitivity analysis of the eigenvalues of the Reissner-Mindlin system. SIAM J. Math. Anal. 47(1), 407–426 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Buoso, D., Lamberti, P.D.: On a classical spectral optimization problem in linear elasticity. In: New Trends in Shape Optimization. Birkhäuser, Basel (2015)

    Google Scholar 

  15. Buoso, D., Provenzano, L.: A few shape optimization results for a biharmonic Steklov problem. J. Differ. Equ. 259(5), 1778–1818 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Buoso, D., Provenzano, L.: On the eigenvalues of a biharmonic Steklov problem. In: Integral Methods in Science and Engineering: Theoretical and Computational Advances. Birkhäuser, Basel (2015)

    Google Scholar 

  17. Burenkov, V.I., Davies, E.B.: Spectral stability of the Neumann Laplacian. J. Differ. Equ. 186(2), 485–508 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Burenkov, V.I., Lamberti, P.D.: Spectral stability of general non-negative selfadjoint operators with applications to Neumann-type operators. J. Differ. Equ. 233(2), 345–379 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Burenkov, V.I., Lamberti, P.D.: Spectral stability of higher order uniformly elliptic operators. In: Sobolev Spaces in Mathematics II. International Mathematical Series (N.Y.), vol. 9, pp. 69–102. Springer, New York (2009)

    Google Scholar 

  20. Burenkov, V.I., Lamberti, P.D.: Sharp spectral stability estimates via the Lebesgue measure of domains for higher order elliptic operators. Rev. Mat. Complut. 25(2), 435–457 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chasman, L.M.: An isoperimetric inequality for fundamental tones of free plates. Commun. Math. Phys. 303(2), 421–449 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chasman, L.M.: An isoperimetric inequality for fundamental tones of free plates with nonzero Poisson’s ratio. Appl. Anal. 95(8), (2016). doi:10.1080/00036811.2015.1068299

    Google Scholar 

  23. Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 1. Wiley, New York (1989)

    Book  MATH  Google Scholar 

  24. Dalmasso, R.: Un problème de symétrie pour une équation biharmonique. Ann. Fac. Sci. Toulouse Math. (5) 11 (3), 45–53 (1990)

    Google Scholar 

  25. Delfour, M.C., Zolésio, J.P.: Shapes and geometries. Analysis, differential calculus, and optimization. In: Advances in Design and Control, 4. Society for Industrial and Applied Mathematics (SIAM). Philadelphia, PA (2001)

    Google Scholar 

  26. Faber, G.: Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt. Sitz. Ber. Bayer. Akad. Wiss. 169–172 (1923)

    Google Scholar 

  27. Gazzola, F., Grunau, H.-C., Sweers, G.: Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains. Lecture Notes in Mathematics. Springer, Berlin (2010)

    MATH  Google Scholar 

  28. Giroire, J., Nédélec, J.-C.: A new system of boundary integral equations for plates with free edges. Math. Methods Appl. Sci. 18(10), 755–772 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  29. Henrot, A.: Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006)

    MATH  Google Scholar 

  30. Henry, D.: Perturbation of the boundary in boundary-value problems of partial differential equations. In: Hale, J., Pereira, A.L. London Mathematical Society Lecture Note Series 318. Cambridge University Press, Cambridge (2005). (With editorial assistance from)

    Google Scholar 

  31. Krahn, E.: Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises. Math. Ann. 94, 97–100 (1924)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kuttler, J.R., Sigillito, V.G.: Inequalities for membrane and Stekloff eigenvalues. J. Math. Anal. Appl. 23, 148–160 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kuttler, J.R.: Remarks on a Stekloff eigenvalue problem. SIAM J. Numer. Anal. 9, 1–5 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lamberti, P.D.: Steklov-type eigenvalues associated with best Sobolev trace constants: domain perturbation and overdetermined systems. Complex Var. Elliptic Equ. 59(3), 309–323 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lamberti, P.D., Lanza de Cristoforis, M.: A real analyticity result for symmetric functions of the eigenvalues of a domain dependent Dirichlet problem for the Laplace operator. J. Nonlinear Convex Anal. 5(1), 19–42 (2004)

    Google Scholar 

  36. Lamberti, P.D., Lanza de Cristoforis, M.: Critical points of the symmetric functions of the eigenvalues of the Laplace operator and overdetermined problems. J. Math. Soc. Jpn. 58(1), 231–245 (2006)

    Google Scholar 

  37. Lamberti, P.D., Lanza de Cristoforis, M.: A real analyticity result for symmetric functions of the eigenvalues of a domain-dependent Neumann problem for the Laplace operator. Mediterr. J. Math. 4(4), 435–449 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  38. Lamberti, P.D., Provenzano, L.: Viewing the Steklov eigenvalues of the Laplace operator as critical Neumann eigenvalues. In: Current Trends in Analysis and Its Applications. Birkhäuser, Basel (2015)

    Google Scholar 

  39. Nadirashvili, N.S.: Rayleigh’s conjecture on the principal frequency of the clamped plate. Arch. Ration. Mech. Anal. 129(1), 1–10 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  40. Ortega, J.H., Zuazua, E.: Generic simplicity of the spectrum and stabilization for a plate equation. SIAM J. Control Optim. 39(5), pp. 1585–1614 (2001). addendum ibid. 42. 5, 1905–1910 (2003)

    Google Scholar 

  41. Parini, E., Stylianou, A.: On the positivity preserving property of hinged plates. SIAM J. Math. Anal. 41(5), 2031–2037 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  42. Provenzano, L.: A note on the Neumann eigenvalues of the biharmonic operator. J. Math. Methods Appl. Sci. submitted

    Google Scholar 

  43. Rayleigh, J.W.S.: The Theory of Sound. Macmillan and Co., London (1877)

    MATH  Google Scholar 

  44. Rellich, F.: Perturbation Theory of Eigenvalue Problems. Gordon and Breach Science Publishers, New York (1969)

    MATH  Google Scholar 

  45. Sweers, G.: A survey on boundary conditions for the biharmonic. Complex Var. Elliptic Equ. 54(2), 79–93 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  46. Talenti, G.: On the first eigenvalue of the clamped plate. Ann. Mat. Pura Appl. (4) 129, 265–280 (1981)

    Google Scholar 

  47. Verchota, G.C.: The biharmonic Neumann problem in Lipschitz domains. Acta Math. 194(2), 217–279 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author is very thankful to Prof. Pier Domenico Lamberti and Dr. Luigi Provenzano for useful comments and discussions. The author has been partially supported by the research project ‘Singular perturbation problems for differential operators’ Progetto di Ateneo of the University of Padova, and by the research project FIR (Futuro in Ricerca) 2013 ‘Geometrical and qualitative aspects of PDE’s’. The author is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Davide Buoso .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Buoso, D. (2016). Analyticity and Criticality Results for the Eigenvalues of the Biharmonic Operator. In: Gazzola, F., Ishige, K., Nitsch, C., Salani, P. (eds) Geometric Properties for Parabolic and Elliptic PDE's. GPPEPDEs 2015. Springer Proceedings in Mathematics & Statistics, vol 176. Springer, Cham. https://doi.org/10.1007/978-3-319-41538-3_5

Download citation

Publish with us

Policies and ethics