Computing the first eigenpair of the p-Laplacian in annuli.
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Publication Date: | 2015 |
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Format: | Article |
Language: | eng |
Source: | Repositório Institucional da UFOP |
Download full: | http://www.repositorio.ufop.br/handle/123456789/7439 https://doi.org/10.1016/j.jmaa.2014.09.016 |
Summary: | We propose a method for computing the first eigenpair of the Dirichlet p-Laplacian, p > 1, in the annulus Ωa,b = {x ∈ RN : a < |x| < b}, N > 1. For each t ∈ (a, b), we use an inverse iteration method to solve two radial eigenvalue problems: one in the annulus Ωa,t, with the corresponding eigenvalue λ−(t) and boundary conditions u(a) = 0 = u (t); and the other in the annulus Ωt,b, with the corresponding eigenvalue λ+(t) and boundary conditions u (t) = 0 = u(b). Next, we adjust the parameter t using a matching procedure to make λ−(t) coincide with λ+(t), thereby obtaining the first eigenvalue λp. Hence, by a simple splicing argument, we obtain the positive, L∞-normalized, radial first eigenfunction up. The matching parameter is the maximum point ρ of up. In order to apply this method, we derive estimates for λ−(t) and λ+(t), and we prove that these functions are monotone and (locally Lipschitz) continuous. Moreover, we derive upper and lower estimates for the maximum point ρ, which we use in the matching procedure, and we also present a direct proof that up converges to the L∞-normalized distance function to the boundary as p → ∞. We also present some numerical results obtained using this method. |
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Ercole, GreyEspírito Santo, Júlio César doMartins, Eder Marinho2017-03-23T15:41:49Z2017-03-23T15:41:49Z2015ERCOLE, G.; ESPÍRITO SANTO, J. C. do.; MARTINS, E. M. Computing the first eigenpair of the p-Laplacian in annuli. Journal of Mathematical Analysis and Applications, v. 422, p. 1277-1307, 2015. Disponível em: <http://www.sciencedirect.com/science/article/pii/S0022247X14008403>. Acesso em: 23 mar. 2017.0022-247Xhttp://www.repositorio.ufop.br/handle/123456789/7439https://doi.org/10.1016/j.jmaa.2014.09.016We propose a method for computing the first eigenpair of the Dirichlet p-Laplacian, p > 1, in the annulus Ωa,b = {x ∈ RN : a < |x| < b}, N > 1. For each t ∈ (a, b), we use an inverse iteration method to solve two radial eigenvalue problems: one in the annulus Ωa,t, with the corresponding eigenvalue λ−(t) and boundary conditions u(a) = 0 = u (t); and the other in the annulus Ωt,b, with the corresponding eigenvalue λ+(t) and boundary conditions u (t) = 0 = u(b). Next, we adjust the parameter t using a matching procedure to make λ−(t) coincide with λ+(t), thereby obtaining the first eigenvalue λp. Hence, by a simple splicing argument, we obtain the positive, L∞-normalized, radial first eigenfunction up. The matching parameter is the maximum point ρ of up. In order to apply this method, we derive estimates for λ−(t) and λ+(t), and we prove that these functions are monotone and (locally Lipschitz) continuous. Moreover, we derive upper and lower estimates for the maximum point ρ, which we use in the matching procedure, and we also present a direct proof that up converges to the L∞-normalized distance function to the boundary as p → ∞. We also present some numerical results obtained using this method.O periódico Journal of Mathematical Analysis and Applications concede permissão para depósito deste artigo no Repositório Institucional da UFOP. Número da licença: 4073071202962.info:eu-repo/semantics/openAccessAnnulusFirst eigenpairInverse iteration methodp-LaplacianComputing the first eigenpair of the p-Laplacian in annuli.info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleengreponame:Repositório Institucional da UFOPinstname:Universidade Federal de Ouro Preto (UFOP)instacron:UFOPLICENSElicense.txtlicense.txttext/plain; charset=utf-8924http://www.repositorio.ufop.br/bitstream/123456789/7439/2/license.txt62604f8d955274beb56c80ce1ee5dcaeMD52ORIGINALARTIGO_ComputingFirstEigenpair.pdfARTIGO_ComputingFirstEigenpair.pdfapplication/pdf1127653http://www.repositorio.ufop.br/bitstream/123456789/7439/1/ARTIGO_ComputingFirstEigenpair.pdf32d588652008181a3f97b34988eedd73MD51123456789/74392019-11-06 08:43:41.927oai:localhost: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ório InstitucionalPUBhttp://www.repositorio.ufop.br/oai/requestrepositorio@ufop.edu.bropendoar:32332019-11-06T13:43:41Repositório Institucional da UFOP - Universidade Federal de Ouro Preto (UFOP)false |
dc.title.pt_BR.fl_str_mv |
Computing the first eigenpair of the p-Laplacian in annuli. |
title |
Computing the first eigenpair of the p-Laplacian in annuli. |
spellingShingle |
Computing the first eigenpair of the p-Laplacian in annuli. Ercole, Grey Annulus First eigenpair Inverse iteration method p-Laplacian |
title_short |
Computing the first eigenpair of the p-Laplacian in annuli. |
title_full |
Computing the first eigenpair of the p-Laplacian in annuli. |
title_fullStr |
Computing the first eigenpair of the p-Laplacian in annuli. |
title_full_unstemmed |
Computing the first eigenpair of the p-Laplacian in annuli. |
title_sort |
Computing the first eigenpair of the p-Laplacian in annuli. |
author |
Ercole, Grey |
author_facet |
Ercole, Grey Espírito Santo, Júlio César do Martins, Eder Marinho |
author_role |
author |
author2 |
Espírito Santo, Júlio César do Martins, Eder Marinho |
author2_role |
author author |
dc.contributor.author.fl_str_mv |
Ercole, Grey Espírito Santo, Júlio César do Martins, Eder Marinho |
dc.subject.por.fl_str_mv |
Annulus First eigenpair Inverse iteration method p-Laplacian |
topic |
Annulus First eigenpair Inverse iteration method p-Laplacian |
description |
We propose a method for computing the first eigenpair of the Dirichlet p-Laplacian, p > 1, in the annulus Ωa,b = {x ∈ RN : a < |x| < b}, N > 1. For each t ∈ (a, b), we use an inverse iteration method to solve two radial eigenvalue problems: one in the annulus Ωa,t, with the corresponding eigenvalue λ−(t) and boundary conditions u(a) = 0 = u (t); and the other in the annulus Ωt,b, with the corresponding eigenvalue λ+(t) and boundary conditions u (t) = 0 = u(b). Next, we adjust the parameter t using a matching procedure to make λ−(t) coincide with λ+(t), thereby obtaining the first eigenvalue λp. Hence, by a simple splicing argument, we obtain the positive, L∞-normalized, radial first eigenfunction up. The matching parameter is the maximum point ρ of up. In order to apply this method, we derive estimates for λ−(t) and λ+(t), and we prove that these functions are monotone and (locally Lipschitz) continuous. Moreover, we derive upper and lower estimates for the maximum point ρ, which we use in the matching procedure, and we also present a direct proof that up converges to the L∞-normalized distance function to the boundary as p → ∞. We also present some numerical results obtained using this method. |
publishDate |
2015 |
dc.date.issued.fl_str_mv |
2015 |
dc.date.accessioned.fl_str_mv |
2017-03-23T15:41:49Z |
dc.date.available.fl_str_mv |
2017-03-23T15:41:49Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.citation.fl_str_mv |
ERCOLE, G.; ESPÍRITO SANTO, J. C. do.; MARTINS, E. M. Computing the first eigenpair of the p-Laplacian in annuli. Journal of Mathematical Analysis and Applications, v. 422, p. 1277-1307, 2015. Disponível em: <http://www.sciencedirect.com/science/article/pii/S0022247X14008403>. Acesso em: 23 mar. 2017. |
dc.identifier.uri.fl_str_mv |
http://www.repositorio.ufop.br/handle/123456789/7439 |
dc.identifier.issn.none.fl_str_mv |
0022-247X |
dc.identifier.doi.none.fl_str_mv |
https://doi.org/10.1016/j.jmaa.2014.09.016 |
identifier_str_mv |
ERCOLE, G.; ESPÍRITO SANTO, J. C. do.; MARTINS, E. M. Computing the first eigenpair of the p-Laplacian in annuli. Journal of Mathematical Analysis and Applications, v. 422, p. 1277-1307, 2015. Disponível em: <http://www.sciencedirect.com/science/article/pii/S0022247X14008403>. Acesso em: 23 mar. 2017. 0022-247X |
url |
http://www.repositorio.ufop.br/handle/123456789/7439 https://doi.org/10.1016/j.jmaa.2014.09.016 |
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eng |
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openAccess |
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