Trudinger-Moser and Adams type inequalities on weighted Sobolev spaces and applications
Autor(a) principal: | |
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Data de Publicação: | 2023 |
Tipo de documento: | Tese |
Idioma: | por |
Título da fonte: | Biblioteca Digital de Teses e Dissertações da UFPB |
Texto Completo: | https://repositorio.ufpb.br/jspui/handle/123456789/28007 |
Resumo: | This thesis studies inequalities and embeddings involving a class of Sobolev spaces with potential weights without assuming any boundary condition. Suppose a Dirichlet boundary condition, those spaces have been extensively studied due to their applicability in radial elliptic problems for operators in great generality which include the p-Laplacian and the k-Hessian operators. In the bounded domains case without any boundary condition, we show a sharp embedding into weighted Lebesgue space Lq θ which generalizes [13, Theorem 1.1] and [22, Theorem 1.1]. Also, we prove sharp Adams-Trudinger-Moser embedding under the full norm and sharp Adams inequality with the Navier boundary condition generalizing [22, Theorem 1.3]. As applications, we conclude that the associated elliptic equations with nonlinearities in both forms of polynomial and exponential growths admit nontrivial solutions. Supposing an unbounded domain, our results provide sharp embeddings into weighted Lebesgue spaces Lq θ and the existence and non-existence of the maximizers for their Trudinger-Moser type inequalities. We also sharpen the maximal integrability by “removing" terms from the exponential series while maintaining the continuity of the embedding. Moreover, we establish the second order Adams’ inequalities with the exact growth condition. |
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Trudinger-Moser and Adams type inequalities on weighted Sobolev spaces and applicationsMatemáticaEspaço de Sobolev com pesoLema radialDesigualdade de Trudinger-MoserDesigualdade de AdamsCondição de crescimento exatoMathematicsWeighted Sobolev spaceRadial lemmaTrudinger-Moser inequalityAdams inequalityExact growth conditionCNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICAThis thesis studies inequalities and embeddings involving a class of Sobolev spaces with potential weights without assuming any boundary condition. Suppose a Dirichlet boundary condition, those spaces have been extensively studied due to their applicability in radial elliptic problems for operators in great generality which include the p-Laplacian and the k-Hessian operators. In the bounded domains case without any boundary condition, we show a sharp embedding into weighted Lebesgue space Lq θ which generalizes [13, Theorem 1.1] and [22, Theorem 1.1]. Also, we prove sharp Adams-Trudinger-Moser embedding under the full norm and sharp Adams inequality with the Navier boundary condition generalizing [22, Theorem 1.3]. As applications, we conclude that the associated elliptic equations with nonlinearities in both forms of polynomial and exponential growths admit nontrivial solutions. Supposing an unbounded domain, our results provide sharp embeddings into weighted Lebesgue spaces Lq θ and the existence and non-existence of the maximizers for their Trudinger-Moser type inequalities. We also sharpen the maximal integrability by “removing" terms from the exponential series while maintaining the continuity of the embedding. Moreover, we establish the second order Adams’ inequalities with the exact growth condition.Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPESEsta tese estuda desigualdades e mergulhos envolvendo uma classe de espaços de Sobolev com pesos na forma de potência sem assumir nenhuma condição de bordo. Assumindo uma condição de Dirichlet no bordo, esses espaços têm sido extensivamente estudados devido a sua aplicabilidade em problemas radiais elípticos com diversos tipos operadores os quais incluem os operadores p-Laplaciano e k-Hessiano. No caso de domínios limitados sem qualquer condição de bordo, mostramos um mergulho ótimo no espaço de Lebesgue com peso Lq θ que generaliza [13, Theorem 1.1] e [22, Theorem 1.1]. Além disso, provamos um mergulho ótimo de Adams-Trudinger- Moser trabalhando com a norma completa e a desigualdade ótima de Adams com a condição de Navier no bordo generalizando [22, Theorem 1.3]. Como aplicações, provamos que as equações elípticas associadas com não linearidades em ambas as formas de crescimento polinomial e exponencial admitem soluções não triviais. Supondo um domínio ilimitado, nossos resultados fornecem mergulhos ótimos em espaços de Lebesgue com peso Lq θ e a existência e não existência dos máximos para suas desigualdades do tipo Trudinger-Moser. Também aprimoramos a integrabilidade máxima “removendo" termos da série exponencial enquanto mantemos a imersão contínua. Além disso, estabelecemos as desigualdades de Adams de segunda ordem com a condição de crescimento exata.Universidade Federal da ParaíbaBrasilMatemáticaPrograma de Pós-Graduação em MatemáticaUFPBdo Ó, João Marcos Bezerrahttp://lattes.cnpq.br/6069135199129029Lu, GuozhenPonciano, Raoní Cabral2023-08-24T11:00:27Z2024-05-302023-08-24T11:00:27Z2023-04-28info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesishttps://repositorio.ufpb.br/jspui/handle/123456789/28007porAttribution-NoDerivs 3.0 Brazilhttp://creativecommons.org/licenses/by-nd/3.0/br/info:eu-repo/semantics/embargoedAccessreponame:Biblioteca Digital de Teses e Dissertações da UFPBinstname:Universidade Federal da Paraíba (UFPB)instacron:UFPB2023-08-25T06:05:31Zoai:repositorio.ufpb.br:123456789/28007Biblioteca Digital de Teses e Dissertaçõeshttps://repositorio.ufpb.br/PUBhttp://tede.biblioteca.ufpb.br:8080/oai/requestdiretoria@ufpb.br|| diretoria@ufpb.bropendoar:2023-08-25T06:05:31Biblioteca Digital de Teses e Dissertações da UFPB - Universidade Federal da Paraíba (UFPB)false |
dc.title.none.fl_str_mv |
Trudinger-Moser and Adams type inequalities on weighted Sobolev spaces and applications |
title |
Trudinger-Moser and Adams type inequalities on weighted Sobolev spaces and applications |
spellingShingle |
Trudinger-Moser and Adams type inequalities on weighted Sobolev spaces and applications Ponciano, Raoní Cabral Matemática Espaço de Sobolev com peso Lema radial Desigualdade de Trudinger-Moser Desigualdade de Adams Condição de crescimento exato Mathematics Weighted Sobolev space Radial lemma Trudinger-Moser inequality Adams inequality Exact growth condition CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA |
title_short |
Trudinger-Moser and Adams type inequalities on weighted Sobolev spaces and applications |
title_full |
Trudinger-Moser and Adams type inequalities on weighted Sobolev spaces and applications |
title_fullStr |
Trudinger-Moser and Adams type inequalities on weighted Sobolev spaces and applications |
title_full_unstemmed |
Trudinger-Moser and Adams type inequalities on weighted Sobolev spaces and applications |
title_sort |
Trudinger-Moser and Adams type inequalities on weighted Sobolev spaces and applications |
author |
Ponciano, Raoní Cabral |
author_facet |
Ponciano, Raoní Cabral |
author_role |
author |
dc.contributor.none.fl_str_mv |
do Ó, João Marcos Bezerra http://lattes.cnpq.br/6069135199129029 Lu, Guozhen |
dc.contributor.author.fl_str_mv |
Ponciano, Raoní Cabral |
dc.subject.por.fl_str_mv |
Matemática Espaço de Sobolev com peso Lema radial Desigualdade de Trudinger-Moser Desigualdade de Adams Condição de crescimento exato Mathematics Weighted Sobolev space Radial lemma Trudinger-Moser inequality Adams inequality Exact growth condition CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA |
topic |
Matemática Espaço de Sobolev com peso Lema radial Desigualdade de Trudinger-Moser Desigualdade de Adams Condição de crescimento exato Mathematics Weighted Sobolev space Radial lemma Trudinger-Moser inequality Adams inequality Exact growth condition CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA |
description |
This thesis studies inequalities and embeddings involving a class of Sobolev spaces with potential weights without assuming any boundary condition. Suppose a Dirichlet boundary condition, those spaces have been extensively studied due to their applicability in radial elliptic problems for operators in great generality which include the p-Laplacian and the k-Hessian operators. In the bounded domains case without any boundary condition, we show a sharp embedding into weighted Lebesgue space Lq θ which generalizes [13, Theorem 1.1] and [22, Theorem 1.1]. Also, we prove sharp Adams-Trudinger-Moser embedding under the full norm and sharp Adams inequality with the Navier boundary condition generalizing [22, Theorem 1.3]. As applications, we conclude that the associated elliptic equations with nonlinearities in both forms of polynomial and exponential growths admit nontrivial solutions. Supposing an unbounded domain, our results provide sharp embeddings into weighted Lebesgue spaces Lq θ and the existence and non-existence of the maximizers for their Trudinger-Moser type inequalities. We also sharpen the maximal integrability by “removing" terms from the exponential series while maintaining the continuity of the embedding. Moreover, we establish the second order Adams’ inequalities with the exact growth condition. |
publishDate |
2023 |
dc.date.none.fl_str_mv |
2023-08-24T11:00:27Z 2023-08-24T11:00:27Z 2023-04-28 2024-05-30 |
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.ufpb.br/jspui/handle/123456789/28007 |
url |
https://repositorio.ufpb.br/jspui/handle/123456789/28007 |
dc.language.iso.fl_str_mv |
por |
language |
por |
dc.rights.driver.fl_str_mv |
Attribution-NoDerivs 3.0 Brazil http://creativecommons.org/licenses/by-nd/3.0/br/ info:eu-repo/semantics/embargoedAccess |
rights_invalid_str_mv |
Attribution-NoDerivs 3.0 Brazil http://creativecommons.org/licenses/by-nd/3.0/br/ |
eu_rights_str_mv |
embargoedAccess |
dc.publisher.none.fl_str_mv |
Universidade Federal da Paraíba Brasil Matemática Programa de Pós-Graduação em Matemática UFPB |
publisher.none.fl_str_mv |
Universidade Federal da Paraíba Brasil Matemática Programa de Pós-Graduação em Matemática UFPB |
dc.source.none.fl_str_mv |
reponame:Biblioteca Digital de Teses e Dissertações da UFPB instname:Universidade Federal da Paraíba (UFPB) instacron:UFPB |
instname_str |
Universidade Federal da Paraíba (UFPB) |
instacron_str |
UFPB |
institution |
UFPB |
reponame_str |
Biblioteca Digital de Teses e Dissertações da UFPB |
collection |
Biblioteca Digital de Teses e Dissertações da UFPB |
repository.name.fl_str_mv |
Biblioteca Digital de Teses e Dissertações da UFPB - Universidade Federal da Paraíba (UFPB) |
repository.mail.fl_str_mv |
diretoria@ufpb.br|| diretoria@ufpb.br |
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1801843013362647040 |