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Numerical Optimization of Eigenvalues of the Dirichlet–Laplace Operator on Domains in Surfaces

  • Straubhaar, Régis 1Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand 11, Case postale 158, 2009 Neuchâtel, Switzerland
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  • Computational Methods in Applied Mathematics. - Walter de Gruyter GmbH. - 2014, vol. 14, no. 3, p. 393-409
Applied Mathematics
Numerical Analysis
Computational Mathematics
English Abstract.Let (M,g) be a smooth and complete surface,
$\Omega \subset M$ be a domain in M, and $\Delta _g$ be the Laplace operator on M. The spectrum of the Dirichlet–Laplace operator on Ω is a sequence $0 < \lambda _1(\Omega ) \le \lambda _2(\Omega ) \le \cdots \nearrow \infty $. A classical question is to ask what is the domain $\Omega ^*$ which minimizes $\lambda _m(\Omega )$ among all domains of a given area, and what is the value of the corresponding $\lambda _m(\Omega _m^*)$. The aim of this article is to present a numerical algorithm using shape optimization and based on the finite element method to find an approximation of a candidate for $\Omega _m^*$. Some verifications with existing numerical results are carried out for the first eigenvalues of domains in ℝ2. Furthermore, some investigations are presented in the two-dimensional sphere to illustrate the case of the positive curvature, in hyperbolic space for the negative curvature and in a hyperboloid for a non-constant curvature.
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  • English
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green
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https://sonar.rero.ch/global/documents/248934
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