Uniqueness in the Freedericksz transition with weak anchoring
Autor(a) principal: | |
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Data de Publicação: | 2009 |
Outros Autores: | , , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10400.2/1468 |
Resumo: | In this paper we consider a boundary value problem for a quasilinear pendulum equation with non-linear boundary conditions that arises in a classical liquid crystals setup, the Freedericksz transition, which is the simplest opto-electronic switch, the result of competition between reorienting effects of an applied electric field and the anchoring to the bounding surfaces. A change of variables transforms the problem into the equation xττ = −f (x) for τ ∈ (−T , T ), with boundary conditions xτ = ±βT f (x) at τ = ∓T , for a convex non-linearity f . By analysing an associated inviscid Burgers’ equation, we prove uniqueness of monotone solutions in the original non-linear boundary value problem. This result has been for many years conjectured in the liquid crystals literature, e.g. in [E.G. Virga, Variational Theories for Liquid Crystals, Appl. Math. Math. Comput., vol. 8, Chapman & Hall, London, 1994] and in [I.W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction, Taylor & Francis, London, 2003]. |
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Uniqueness in the Freedericksz transition with weak anchoringFreedericksz transitionBurgers’ equationConvexityNon-linear boundary value problemsUniqueness of solutionsIn this paper we consider a boundary value problem for a quasilinear pendulum equation with non-linear boundary conditions that arises in a classical liquid crystals setup, the Freedericksz transition, which is the simplest opto-electronic switch, the result of competition between reorienting effects of an applied electric field and the anchoring to the bounding surfaces. A change of variables transforms the problem into the equation xττ = −f (x) for τ ∈ (−T , T ), with boundary conditions xτ = ±βT f (x) at τ = ∓T , for a convex non-linearity f . By analysing an associated inviscid Burgers’ equation, we prove uniqueness of monotone solutions in the original non-linear boundary value problem. This result has been for many years conjectured in the liquid crystals literature, e.g. in [E.G. Virga, Variational Theories for Liquid Crystals, Appl. Math. Math. Comput., vol. 8, Chapman & Hall, London, 1994] and in [I.W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction, Taylor & Francis, London, 2003].ElsevierRepositório AbertoCosta, Fernando Pestana daGrinfeld, MichaelMottram, Nigel J.Pinto, João Teixeira2010-05-14T12:07:33Z2009-02-112009-02-11T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10400.2/1468engCosta, Fernando Pestana da [et al.] - Uniqueness in the Freedericksz transition with weak anchoring. "Journal of Differential Equations" [Em linha]. ISSN 0022-0396. Nº 246 (2009), p. 2590-26000022-0396info:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2023-11-16T15:14:06Zoai:repositorioaberto.uab.pt:10400.2/1468Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T22:43:18.223211Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
Uniqueness in the Freedericksz transition with weak anchoring |
title |
Uniqueness in the Freedericksz transition with weak anchoring |
spellingShingle |
Uniqueness in the Freedericksz transition with weak anchoring Costa, Fernando Pestana da Freedericksz transition Burgers’ equation Convexity Non-linear boundary value problems Uniqueness of solutions |
title_short |
Uniqueness in the Freedericksz transition with weak anchoring |
title_full |
Uniqueness in the Freedericksz transition with weak anchoring |
title_fullStr |
Uniqueness in the Freedericksz transition with weak anchoring |
title_full_unstemmed |
Uniqueness in the Freedericksz transition with weak anchoring |
title_sort |
Uniqueness in the Freedericksz transition with weak anchoring |
author |
Costa, Fernando Pestana da |
author_facet |
Costa, Fernando Pestana da Grinfeld, Michael Mottram, Nigel J. Pinto, João Teixeira |
author_role |
author |
author2 |
Grinfeld, Michael Mottram, Nigel J. Pinto, João Teixeira |
author2_role |
author author author |
dc.contributor.none.fl_str_mv |
Repositório Aberto |
dc.contributor.author.fl_str_mv |
Costa, Fernando Pestana da Grinfeld, Michael Mottram, Nigel J. Pinto, João Teixeira |
dc.subject.por.fl_str_mv |
Freedericksz transition Burgers’ equation Convexity Non-linear boundary value problems Uniqueness of solutions |
topic |
Freedericksz transition Burgers’ equation Convexity Non-linear boundary value problems Uniqueness of solutions |
description |
In this paper we consider a boundary value problem for a quasilinear pendulum equation with non-linear boundary conditions that arises in a classical liquid crystals setup, the Freedericksz transition, which is the simplest opto-electronic switch, the result of competition between reorienting effects of an applied electric field and the anchoring to the bounding surfaces. A change of variables transforms the problem into the equation xττ = −f (x) for τ ∈ (−T , T ), with boundary conditions xτ = ±βT f (x) at τ = ∓T , for a convex non-linearity f . By analysing an associated inviscid Burgers’ equation, we prove uniqueness of monotone solutions in the original non-linear boundary value problem. This result has been for many years conjectured in the liquid crystals literature, e.g. in [E.G. Virga, Variational Theories for Liquid Crystals, Appl. Math. Math. Comput., vol. 8, Chapman & Hall, London, 1994] and in [I.W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction, Taylor & Francis, London, 2003]. |
publishDate |
2009 |
dc.date.none.fl_str_mv |
2009-02-11 2009-02-11T00:00:00Z 2010-05-14T12:07:33Z |
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.uri.fl_str_mv |
http://hdl.handle.net/10400.2/1468 |
url |
http://hdl.handle.net/10400.2/1468 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Costa, Fernando Pestana da [et al.] - Uniqueness in the Freedericksz transition with weak anchoring. "Journal of Differential Equations" [Em linha]. ISSN 0022-0396. Nº 246 (2009), p. 2590-2600 0022-0396 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.publisher.none.fl_str_mv |
Elsevier |
publisher.none.fl_str_mv |
Elsevier |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
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Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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RCAAP |
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RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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1799134999043112960 |