Annali di Matematica Pura ed Applicata

, Volume 186, Issue 2, pp 341–358 | Cite as

Optimization of the first Steklov eigenvalue in domains with holes: a shape derivative approach

  • Julián Fernández Bonder
  • Pablo Groisman
  • Julio D. Rossi


The best Sobolev trace constant is given by the first eigenvalue of a Steklov-like problem. We deal with minimizers of the Rayleigh quotient ‖u2 H 1 (Ω) 2/‖u2 L 2 (∂Ω) for functions that vanish in a subset A⊂ Ω, which we call the hole. We look for holes that minimize the best Sobolev trace constant among subsets of Ω with prescribed volume. First, we find a formula for the first variation of the first eigenvalue with respect to the hole. As a consequence of this formula, we prove that when Ω is a ball the symmetric hole (a centered ball) is critical when we consider deformations that preserves volume but is not optimal. Finally, we prove that by the Finite Element Method we can approximate the optimal configuration and, by means of the shape derivative, we design an algorithm to compute the discrete optimal holes.


Steklov eigenvalues Sobolev trace embedding Shape derivative 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aubin, T.: équations différentielles non linéaires et le problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. 55, 269–296 (1976)MathSciNetGoogle Scholar
  2. 2.
    Biezuner, R.J.: Best constants in Sobolev trace inequalities. Nonlinear Anal. 54, 575–589 (2003)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Cherkaev, A., Cherkaeva, E.: Optimal design for uncertain loading condition. Ser. Adv. Math. Appl. Sci., 50. Homogenization, pp. 193–213. World Science Publishing, River Edge, NJ (1999)Google Scholar
  4. 4.
    Ciarlet, P.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978)Google Scholar
  5. 5.
    del Pino, M., Flores, C.: Asymptotic behavior of best constants and extremals for trace embeddings in expanding domains. Comm. Partial Differ. Equations 26(11/12), 2189–2210 (2001)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Druet, O., Hebey, E.: The AB program in geometric analysis: sharp Sobolev inequalities and related problems. Memoirs Am. Math. Soc. 160(761) (2002)Google Scholar
  7. 7.
    Escobar, J.F.: Sharp constant in a Sobolev trace inequality. Indiana Math. J. 37(3), 687–698 (1988)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Fernández Bonder, J., Lami Dozo, E., Rossi, J.D.: Symmetry properties for the extremals of the Sobolev trace embedding. Ann. Inst. H. Poincaré. Anal. Non Linéaire 21(6), 795–805 (2004)CrossRefGoogle Scholar
  9. 9.
    Fernández Bonder, J., Rossi, J.D.: Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains. Comm. Pure Appl. Anal. 1(3), 359–378 (2002)Google Scholar
  10. 10.
    Fernández Bonder, J., Rossi, J.D.: On the existence of extremals for the Sobolev trace embedding theorem with critical exponent. Bull. London Math. Soc. 37(1), 119–125 (2005)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Fernández Bonder, J., Rossi, J.D., Wolanski, N.: Behavior of the best Sobolev trace constant and extremals in domains with holes. To appear in Bull. Sci. Math.Google Scholar
  12. 12.
    Fernández Bonder, J., Rossi, J.D., Wolanski, N.: Regularity of the free boundary in an optimization problem related to the best Sobolev trace constant. SIAM J. Control Optim. 44(5), 1614–1635 (2005)Google Scholar
  13. 13.
    Henrot, A.: Minimization problems for eigenvalues of the Laplacian. J. Evol. Equ. 3(3), 443–461 (2003)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Henrot, A., Pierre, M.: Optimization de forme. Collection: Mathématiques et Applications. Springer, vol. 48 (2005), ISBN: 3-540-26211-3Google Scholar
  15. 15.
    Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin Heidelberg New York (1995)Google Scholar
  16. 16.
    Lami Dozo, E., Torne, O.: Symmetry and symmetry breaking for minimizers in the trace inequality Commun. Contemp. Math. 7(6), 727–746 (2005)Google Scholar
  17. 17.
    Li, Y., Zhu, M.: Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries. Comm. Pure Appl. Math. 50, 449–487 (1997)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Martinez, S., Rossi, J.D.: Isolation and simplicity for the first eigenvalue of the p-Laplacian with a nonlinear boundary condition. Abst. Appl. Anal. 7(5), 287–293 (2002)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Pironeau, O.: Optimal Shape Design for Elliptic Systems. Springer Berlin Heidelberg New York (1984)Google Scholar
  20. 20.
    Schwartz, L.: Cours d'analyse. 1 et 2. Second edition. Hermann, Paris (1981)Google Scholar
  21. 21.
    Sperner, E.: Spherical symmetrization and eigenvalue estimates. Math. Z. 176, 75–86 (1981)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Steklov, M.W.: Sur les problèmes fondamentaux en physique mathématique. Ann. Sci. Ecole Norm. Sup. 19, 455–490 (1902)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Julián Fernández Bonder
    • 1
  • Pablo Groisman
    • 2
  • Julio D. Rossi
    • 3
  1. 1.Departamento de Matemática, FCEyNUniversidad de Buenos AiresBuenos AiresArgentina
  2. 2.Instituto de Cálculo, FCEyNUniversidad de Buenos AiresBuenos AiresArgentina
  3. 3.Consejo Superior de Investigaciones Científicas (CSIC)MadridSpain

Personalised recommendations