Applied Mathematics & Optimization

, Volume 75, Issue 1, pp 1–25 | Cite as

The Steklov Spectrum on Moving Domains

Article
First Online:

Abstract

We study the continuity of the Steklov spectrum on variable domains with respect to the Hausdorff convergence. A key point of the article is understanding the behaviour of the traces of Sobolev functions on moving boundaries of sets satisfying an uniform geometric condition. As a consequence, we are able to prove existence results for shape optimization problems regarding the Steklov spectrum in the family of sets satisfying a \(\varepsilon \)-cone condition and in the family of convex sets.

Keywords

Eigenvalues Steklov spectrum Shape optimization 

Mathematics Subject Classification

49Q10 35P15 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

The author wishes to thank Dorin Bucur for the inspiring discussions and suggestions regarding this work.

References

  1. 1.
    Brock, F.: An isoperimetric inequality for eigenvalues of the Stekloff problem. ZAMM Z. Angew. Math. Mech. 81(1), 69–71 (2001)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bucur, D.: How to prove existence in shape optimization. Control Cybernet. 34(1), 103–116 (2005)MathSciNetMATHGoogle Scholar
  3. 3.
    Bucur, D., Buttazzo, G.: Variational Methods in Shape Optimization Problems. Progress in Nonlinear Differential Equations and Their Applications, vol. 65. Birkhäuser Boston, Inc., Boston (2005)MATHGoogle Scholar
  4. 4.
    Bucur, D., Gazzola, F.: The first biharmonic Steklov eigenvalue: positivity preserving and shape optimization. Milan J. Math. 79(1), 247–258 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Colbois, B., El Soufi, A., Girouard, A.: Isoperimetric control of the Steklov spectrum. J. Funct. Anal. 261(5), 1384–1399 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Evans, L.C.: Weak Convergence Methods for Nonlinear Partial Differential Equations, Volume 74 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC. American Mathematical Society, Providence (1990)Google Scholar
  7. 7.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)MATHGoogle Scholar
  8. 8.
    Girouard, A., Polterovich, I.: On the Hersch-Payne-Schiffer estimates for the eigenvalues of the Steklov problem. Funktsional. Anal. i Prilozhen. 44(2), 33–47 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Girouard, A., Polterovich, I.: Shape optimization for low Neumann and Steklov eigenvalues. Math. Methods Appl. Sci. 33(4), 501–516 (2010)MathSciNetMATHGoogle Scholar
  10. 10.
    Henrot, A.: Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006)MATHGoogle Scholar
  11. 11.
    Henrot, A., Pierre, M.: Variation et optimisation de formes, Volume 48 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer, Berlin (2005). Une analyse géométrique. [A geometric analysis]Google Scholar
  12. 12.
    Maggi, F.: Sets of Finite Perimeter and Geometric Variational Problems, Volume 135 of Cambridge Studies in Advanced Mathematics. An Introduction to Geometric Measure Theory. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  13. 13.
    Royden, H.L.: Real Analysis. The Macmillan Co. and Collier-Macmillan Ltd., New York and London (1963)MATHGoogle Scholar
  14. 14.
    Tiba, D.: A property of Sobolev spaces and existence in optimal design. Appl. Math. Optim. 47(1), 45–58 (2003)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Weinstock, R.: Inequalities for a classical eigenvalue problem. J. Rational Mech. Anal. 3, 745–753 (1954)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques UMR CNRS 5127Université de SavoieLe Bourget du LacFrance

Personalised recommendations