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Numerical Optimization of Low Eigenvalues of the Dirichlet and Neumann Laplacians

Author

Listed:
  • Pedro R. S. Antunes

    (Universidade Lusófona de Humanidades e Tecnologias
    Complexo Interdisciplinar)

  • Pedro Freitas

    (Complexo Interdisciplinar
    TU Lisbon)

Abstract

We perform a numerical optimization of the first ten nontrivial eigenvalues of the Neumann Laplacian for planar Euclidean domains. The optimization procedure is done via a gradient method, while the computation of the eigenvalues themselves is done by means of an efficient meshless numerical method which allows for the computation of the eigenvalues for large numbers of domains within a reasonable time frame. The Dirichlet problem, previously studied by Oudet using a different numerical method, is also studied and we obtain similar (but improved) results for a larger number of eigenvalues. These results reveal an underlying structure to the optimizers regarding symmetry and connectedness, for instance, but also show that there are exceptions to these preventing general results from holding.

Suggested Citation

  • Pedro R. S. Antunes & Pedro Freitas, 2012. "Numerical Optimization of Low Eigenvalues of the Dirichlet and Neumann Laplacians," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 235-257, July.
    • Handle: RePEc:spr:joptap:v:154:y:2012:i:1:d:10.1007_s10957-011-9983-3
    DOI: 10.1007/s10957-011-9983-3
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    Cited by:

    1. Meizhi Qian & Shengfeng Zhu, 2022. "A level set method for Laplacian eigenvalue optimization subject to geometric constraints," Computational Optimization and Applications, Springer, vol. 82(2), pages 499-524, June.
    2. Pedro R. S. Antunes & Beniamin Bogosel, 2022. "Parametric shape optimization using the support function," Computational Optimization and Applications, Springer, vol. 82(1), pages 107-138, May.
    3. Shengfeng Zhu, 2018. "Effective Shape Optimization of Laplace Eigenvalue Problems Using Domain Expressions of Eulerian Derivatives," Journal of Optimization Theory and Applications, Springer, vol. 176(1), pages 17-34, January.

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