SpringerLink
Log in

Optimisation of Eigenvalues of the Dirichlet Laplacian with a Surface Area Restriction

  • Published:
Applied Mathematics & Optimization Aims and scope Submit manuscript

Abstract

We perform a numerical optimisation of the low frequencies of the Dirichlet Laplacian with perimeter and surface area restrictions, in two and 3-dimensions, respectively. In the former case, we handle the first 50 eigenvalues and measure the rate at which the corresponding optimisers approach the disk, while in the latter we optimise the first twenty eigenvalues. We derive theoretical compatibility conditions which must be satisfied by a sequence of optimisers and test our numerical results against these. We also consider the cases of rectangles with a fixed perimeter and parallelepipeds with a surface restriction for which we compute the first \(10^7\) and \(10^6\) optimal eigenvalues, respectively. In this context, we prove convergence to the cube in any dimensions and compare the numerical results with our theoretical estimates for the rate of convergence.

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

Access this article

Log in via an institution

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

Price includes VAT (Canada)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Antunes, P.R.S., Freitas, P.: Numerical optimisation of low eigenvalues of the Dirichlet and Neumann Laplacians. J. Optim. Theory Appl. 154, 235–257 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Oudet, E.: Numerical minimization of eigenmodes of a membrane with respect to the domain. ESAIM Control Optim. Calc. Var. 10, 315–330 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Antunes, P.R.S., Freitas, P.: Optimal spectral rectangles and lattice ellipses. Proc. R. Soc. Lond. A 469, 20120492 (2013). doi:10.1098/rspa.2012.0492

    Article  MathSciNet  Google Scholar 

  4. Antunes, P.R.S., Freitas, P., Kennedy, J.B.: Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin Laplacian. ESAIM: Control Optim. Calc. Var. 19, 438–459 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bucur, D., Freitas, P.: Asymptotic behaviour of optimal spectral planar domains with fixed perimeter. J. Math. Phys. 54, 053504 (2013). doi:10.1063/1.4803140

    Article  MathSciNet  MATH  Google Scholar 

  6. Colbois, B., EI Soufi, A.: Extremal eigenvalues of the Laplacian on Euclidean domains and closed surfaces. Math. Z. 278(1—-2), 529–549 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. De Philippis, G., Velichkov, B.: Existence and regularity of minimizers for some spectral functionals with perimeter constraint. Appl. Math. Optim. 69(2), 199–231 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bucur, D., Buttazzo, G., Henrot, A.: Minimization of \(\lambda _2(\Omega )\) with a perimeter constraint. Indiana Univ. Math. J. 5(6), 2709–2728 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kao, C.-Y., Osting, B.: Minimal convex combinations of sequential Laplace–Dirichlet eigenvalues. SIAM J. Sci. Comput. 35(3), B731–B750 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Levitin, M., Yagudin, R.: Range of the first three eigenvalues of the planar Dirichlet Laplacian. LMS J. Comput. Math. 6, 1–17 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Berger, A.: Optimisation du spectre du Laplacien avec conditions de Dirichlet et Neumann dans \(R^2\) et \(R^3\), PhD thesis, Université de Grenoble et Université de Neuchâtel (2015)

  12. Bogosel B., Oudet E.: Qualitative and numerical analysis of a spectral problem with perimeter constraint, preprint (2015)

  13. Lewis A.S., Overton M.L.: Nonsmooth optimization via BFGS (preprint). Available at http://www.optimization-online.org/DB_FILE/2008/12/2172.pdf

  14. Alves, C.J.S., Antunes, P.R.S.: The method of fundamental solutions applied to some inverse eigenproblems. SIAM J. Sci. Comput. 25(3), A1689–A1708 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Antunes, P.R.S.: Numerical calculation of eigensolutions of 3D shapes using the method of fundamental solutions. Numer. Methods Partial Differ. Equs. 27(6), 1525–1550 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Pólya, G.: On the eigenvalues of vibrating membranes. Proc. Lond. Math. Soc. 11, 419–433 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  17. Urakawa, H.: Lower bounds for the eigenvalues of the fixed vibrating membrane problems. Tohoku Math. J. 36, 185–189 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  18. Safarov, Y., Vassiliev, D.: The Asymptotic Distribution of Eigenvalues of Partial Differential Operators. Series translations of mathematical monographs. American Mathematical Society, Providence (1997)

    MATH  Google Scholar 

Download references

Acknowledgments

Pedro R. S. Antunes was partially supported by FCT, Portugal, through the program “Investigador FCT” with reference IF/00177/2013.

Author information

Authors and Affiliations

  1. Group of Mathematical Physics, Universidade de Lisboa, Complexo Interdisciplinar, Av. Prof. Gama Pinto 2, 1649-003, Lisbon, Portugal

    Pedro R. S. Antunes & Pedro Freitas

  2. Department of Mathematics, Faculdade de Motricidade Humana, Universidade de Lisboa, Lisbon, Portugal

    Pedro Freitas

Authors
  1. Pedro R. S. Antunes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pedro Freitas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pedro R. S. Antunes.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Antunes, P.R.S., Freitas, P. Optimisation of Eigenvalues of the Dirichlet Laplacian with a Surface Area Restriction. Appl Math Optim 73, 313–328 (2016). https://doi.org/10.1007/s00245-015-9304-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-015-9304-6

Keywords

Mathematics Subject Classification

Search

Navigation