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Optimal partitions for first eigenvalues of the Laplace operator
Author(s) -
Bozorgnia Farid
Publication year - 2015
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.21927
Subject(s) - mathematics , eigenvalues and eigenvectors , laplace operator , eigenfunction , operator (biology) , bounded function , rate of convergence , convergence (economics) , dirichlet eigenvalue , dirichlet boundary condition , boundary value problem , mathematical analysis , dirichlet's principle , biochemistry , physics , chemistry , channel (broadcasting) , repressor , quantum mechanics , economic growth , transcription factor , electrical engineering , economics , gene , engineering
Given a bounded open set Ω ⊂ ℝ 2 , we present numerical approximations for two problems related to minimal partitioning of the first eigenvalues of the Dirichlet Laplacian. The first problem is about minimizing the sum of first eigenvalues of the Dirichlet Laplacian. This partitioning problem arises as a steady state of a reaction‐diffusion process. To do this, a new idea to approximate the second eigenfunction and second eigenvalue is presented. We use the qualitative properties of the minimization problem to construct a numerical algorithm to approximate optimal configurations. A rigorous analysis to show the convergence and the rate of convergence is given. Moreover, we discuss the numerical implementation of the resulting approach and present computational tests confirming the expected asymptotic behavior of optimal partitions with large numbers of partitions. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 923–949, 2015