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Optimization of the First Dirichlet Laplacian Eigenvalue with Respect to a Union of Balls

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Autor(es):
Birgin, E. G. ; Fernandez, L. ; Haeser, G. ; Laurain, A.
Número total de Autores: 4
Tipo de documento: Artigo Científico
Fonte: JOURNAL OF GEOMETRIC ANALYSIS; v. 33, n. 6, p. 28-pg., 2023-06-01.
Resumo

The problem of minimizing the first eigenvalue of the Dirichlet Laplacian with respect to a union of m balls with fixed identical radii and variable centers in the plane is investigated in the present work. The existence of a minimizer is shown and the shape sensitivity analysis of the eigenvalue with respect to the centers' positions is presented. With this tool, the derivative of the eigenvalue is computed and used in a numerical algorithm to determine candidates for minimizers. Candidates are also constructed by hand based on regular polygons. Numerical solutions contribute in at least three aspects. They corroborate the idea that some of the candidates based on regular polygons might be optimal. They also suggest alternative regular patterns that improve solutions associated with regular polygons. Lastly and most importantly, they delivered better quality solutions that do not follow any apparent pattern. Overall, for low values of m, candidates for minimizers of the eigenvalue are proposed and their geometrical properties as well as the appearance of regular patterns formed by the centers are discussed. (AU)

Processo FAPESP: 18/24293-0 - Métodos computacionais de otimização
Beneficiário:Sandra Augusta Santos
Modalidade de apoio: Auxílio à Pesquisa - Temático
Processo FAPESP: 13/07375-0 - CeMEAI - Centro de Ciências Matemáticas Aplicadas à Indústria
Beneficiário:Francisco Louzada Neto
Modalidade de apoio: Auxílio à Pesquisa - Centros de Pesquisa, Inovação e Difusão - CEPIDs
Processo FAPESP: 16/01860-1 - Problemas de corte, empacotamento, dimensionamento de lotes, programação da produção, roteamento, localização e suas integrações em contextos industriais e logísticos
Beneficiário:Reinaldo Morabito Neto
Modalidade de apoio: Auxílio à Pesquisa - Temático