On a class of singular elliptic equation arising in MEMS modeling
Autor(a) principal: | |
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Data de Publicação: | 2014 |
Tipo de documento: | Tese |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UFPE |
Texto Completo: | https://repositorio.ufpe.br/handle/123456789/29833 |
Resumo: | Motivated by recent works on the study of the equations that model the electrostatic MEMS devices, we study the radial solutions of some quasilinear elliptic equations with nonlinearity of inverse square type. According to the choice of the parameters on the problem, the differential operator which we are dealing with corresponds to the radial form of p-Laplacian (p > 1) and k-Hessian. We prove the existence of an extremal parameter λ* > 0 such that for λ ∈ (0, λ*) there exists a minimal solution uλ and for λ > λ* there is no solution of any considered kind. Via Shooting Method, we prove uniqueness of solutions for λ close to 0. We also study the behavior of the minimal branch of solutions. Concerning the case λ = λ*, we prove uniqueness of solutions and present a regularity result. In addition, we present conditions over which we can assert regularity for the critical solution with respect to the parameter λ for the existence of solutions. Moreover, we prove that whenever the critical solution is regular, there exists other solutions of mountain pass type for λ close to λ*. |
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SILVA, Esteban Pereira dahttp://lattes.cnpq.br/8085933568620489http://lattes.cnpq.br/6069135199129029DO Ó, João Marcos Bezerra2019-03-21T11:57:13Z2019-03-21T11:57:13Z2014-12-12https://repositorio.ufpe.br/handle/123456789/29833Motivated by recent works on the study of the equations that model the electrostatic MEMS devices, we study the radial solutions of some quasilinear elliptic equations with nonlinearity of inverse square type. According to the choice of the parameters on the problem, the differential operator which we are dealing with corresponds to the radial form of p-Laplacian (p > 1) and k-Hessian. We prove the existence of an extremal parameter λ* > 0 such that for λ ∈ (0, λ*) there exists a minimal solution uλ and for λ > λ* there is no solution of any considered kind. Via Shooting Method, we prove uniqueness of solutions for λ close to 0. We also study the behavior of the minimal branch of solutions. Concerning the case λ = λ*, we prove uniqueness of solutions and present a regularity result. In addition, we present conditions over which we can assert regularity for the critical solution with respect to the parameter λ for the existence of solutions. Moreover, we prove that whenever the critical solution is regular, there exists other solutions of mountain pass type for λ close to λ*.CNPqMotivados pela recente atenção dada ao estudo das equações que modelam MEMS eletrostáticos, estudamos as soluções radiais de uma classe de equações com não linearidade do tipo inverso do quadrado. De acordo com a escolha dos parâmetros envolvidos no problema, o operador diferencial com o qual lidamos corresponde à forma radial do p-laplaciano (p > 1) e k-hessiano. Provamos a existência de um parâmetro extremal λ* > 0 tal que, para λ ∈ (0, λ*), existe uma solução minimal não singular para o problema. Para λ > λ* não há solução de nenhum tipo considerado. Estudamos também o comportamento do ramo de soluções minimais e exibimos um método para aproximação numérica destas soluções. No que se refere ao caso λ = λ*, provamos unicidade de solução e apresentamos um resultado de regularidade. Além disso, apresentamos condições sobre as quais é possível garantir a regularidade da solução crítica (λ = λ*). Provamos também que sempre que a solução crítica for regular, existe uma outra solução do tipo passo da montanha para λ perto de λ*.engUniversidade Federal de PernambucoPrograma de Pos Graduacao em MatematicaUFPEBrasilAttribution-NonCommercial-NoDerivs 3.0 Brazilhttp://creativecommons.org/licenses/by-nc-nd/3.0/br/info:eu-repo/semantics/openAccessAnálise matemáticaEquações diferenciais ordináriasEquações diferenciais não linearesOn a class of singular elliptic equation arising in MEMS modelinginfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisdoutoradoreponame:Repositório Institucional da UFPEinstname:Universidade Federal de Pernambuco (UFPE)instacron:UFPETHUMBNAILTESE Esteban Pereira da Silva.pdf.jpgTESE Esteban Pereira da Silva.pdf.jpgGenerated Thumbnailimage/jpeg1367https://repositorio.ufpe.br/bitstream/123456789/29833/6/TESE%20Esteban%20Pereira%20da%20Silva.pdf.jpgf82837445833631f12033e18e990ee8fMD56ORIGINALTESE Esteban Pereira da Silva.pdfTESE Esteban Pereira da Silva.pdfapplication/pdf1079043https://repositorio.ufpe.br/bitstream/123456789/29833/1/TESE%20Esteban%20Pereira%20da%20Silva.pdfe504003ea12681e4a4c9bb6a3c23f598MD51LICENSElicense.txtlicense.txttext/plain; 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dc.title.pt_BR.fl_str_mv |
On a class of singular elliptic equation arising in MEMS modeling |
title |
On a class of singular elliptic equation arising in MEMS modeling |
spellingShingle |
On a class of singular elliptic equation arising in MEMS modeling SILVA, Esteban Pereira da Análise matemática Equações diferenciais ordinárias Equações diferenciais não lineares |
title_short |
On a class of singular elliptic equation arising in MEMS modeling |
title_full |
On a class of singular elliptic equation arising in MEMS modeling |
title_fullStr |
On a class of singular elliptic equation arising in MEMS modeling |
title_full_unstemmed |
On a class of singular elliptic equation arising in MEMS modeling |
title_sort |
On a class of singular elliptic equation arising in MEMS modeling |
author |
SILVA, Esteban Pereira da |
author_facet |
SILVA, Esteban Pereira da |
author_role |
author |
dc.contributor.authorLattes.pt_BR.fl_str_mv |
http://lattes.cnpq.br/8085933568620489 |
dc.contributor.advisorLattes.pt_BR.fl_str_mv |
http://lattes.cnpq.br/6069135199129029 |
dc.contributor.author.fl_str_mv |
SILVA, Esteban Pereira da |
dc.contributor.advisor1.fl_str_mv |
DO Ó, João Marcos Bezerra |
contributor_str_mv |
DO Ó, João Marcos Bezerra |
dc.subject.por.fl_str_mv |
Análise matemática Equações diferenciais ordinárias Equações diferenciais não lineares |
topic |
Análise matemática Equações diferenciais ordinárias Equações diferenciais não lineares |
description |
Motivated by recent works on the study of the equations that model the electrostatic MEMS devices, we study the radial solutions of some quasilinear elliptic equations with nonlinearity of inverse square type. According to the choice of the parameters on the problem, the differential operator which we are dealing with corresponds to the radial form of p-Laplacian (p > 1) and k-Hessian. We prove the existence of an extremal parameter λ* > 0 such that for λ ∈ (0, λ*) there exists a minimal solution uλ and for λ > λ* there is no solution of any considered kind. Via Shooting Method, we prove uniqueness of solutions for λ close to 0. We also study the behavior of the minimal branch of solutions. Concerning the case λ = λ*, we prove uniqueness of solutions and present a regularity result. In addition, we present conditions over which we can assert regularity for the critical solution with respect to the parameter λ for the existence of solutions. Moreover, we prove that whenever the critical solution is regular, there exists other solutions of mountain pass type for λ close to λ*. |
publishDate |
2014 |
dc.date.issued.fl_str_mv |
2014-12-12 |
dc.date.accessioned.fl_str_mv |
2019-03-21T11:57:13Z |
dc.date.available.fl_str_mv |
2019-03-21T11:57:13Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/doctoralThesis |
format |
doctoralThesis |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
https://repositorio.ufpe.br/handle/123456789/29833 |
url |
https://repositorio.ufpe.br/handle/123456789/29833 |
dc.language.iso.fl_str_mv |
eng |
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eng |
dc.rights.driver.fl_str_mv |
Attribution-NonCommercial-NoDerivs 3.0 Brazil http://creativecommons.org/licenses/by-nc-nd/3.0/br/ info:eu-repo/semantics/openAccess |
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Attribution-NonCommercial-NoDerivs 3.0 Brazil http://creativecommons.org/licenses/by-nc-nd/3.0/br/ |
eu_rights_str_mv |
openAccess |
dc.publisher.none.fl_str_mv |
Universidade Federal de Pernambuco |
dc.publisher.program.fl_str_mv |
Programa de Pos Graduacao em Matematica |
dc.publisher.initials.fl_str_mv |
UFPE |
dc.publisher.country.fl_str_mv |
Brasil |
publisher.none.fl_str_mv |
Universidade Federal de Pernambuco |
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reponame:Repositório Institucional da UFPE instname:Universidade Federal de Pernambuco (UFPE) instacron:UFPE |
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UFPE |
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UFPE |
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Repositório Institucional da UFPE |
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Repositório Institucional da UFPE |
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