Smallness and cancellation in some elliptic systems with measure data
Autor(a) principal: | |
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Data de Publicação: | 2018 |
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/10773/23634 |
Resumo: | In a bounded open subset Ω ⊂ Rn, we study Dirichlet problems with elliptic systems, involving a finite Radon measure μ on Rn with values into RN , defined by { −div A(x, u(x), Du(x)) = μ in Ω, u = 0 on ∂Ω, where Aα i (x, y, ξ) = N∑ β=1 n∑ j=1 aα,β i,j (x, y) ξβ j with α ∈ {1, . . . , N } the equation index. We prove the existence of a (distributional) solution u : Ω → RN , obtained as the limit of approximations, by assuming: (i) that coefficients aα,β i,j are bounded Carathéodory functions; (ii) ellipticity of the diagonal coefficients aα,α i,j ; and (iii) smallness of the quadratic form associated to the off-diagonal coefficients aα,β i,j (i.e. α = β) verifying a r-staircase support condition with r > 0. Such a smallness condition is satisfied, for instance, in each one of these cases: (a) aα,β i,j = −aβ,α j,i (skew-symmetry); (b) |aα,β i,j | is small; (c) aα,β i,j may be decomposed into two parts, the first enjoying skew-symmetry and the second being small in absolute value. We give an example that satisfies our hypotheses but does not satisfy assumptions introduced in previous works. A Brezis’s type nonexistence result is also given for general (smooth) elliptic-hyperbolic systems. |
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Smallness and cancellation in some elliptic systems with measure dataEllipticSystemExistenceMeasureSolutionIn a bounded open subset Ω ⊂ Rn, we study Dirichlet problems with elliptic systems, involving a finite Radon measure μ on Rn with values into RN , defined by { −div A(x, u(x), Du(x)) = μ in Ω, u = 0 on ∂Ω, where Aα i (x, y, ξ) = N∑ β=1 n∑ j=1 aα,β i,j (x, y) ξβ j with α ∈ {1, . . . , N } the equation index. We prove the existence of a (distributional) solution u : Ω → RN , obtained as the limit of approximations, by assuming: (i) that coefficients aα,β i,j are bounded Carathéodory functions; (ii) ellipticity of the diagonal coefficients aα,α i,j ; and (iii) smallness of the quadratic form associated to the off-diagonal coefficients aα,β i,j (i.e. α = β) verifying a r-staircase support condition with r > 0. Such a smallness condition is satisfied, for instance, in each one of these cases: (a) aα,β i,j = −aβ,α j,i (skew-symmetry); (b) |aα,β i,j | is small; (c) aα,β i,j may be decomposed into two parts, the first enjoying skew-symmetry and the second being small in absolute value. We give an example that satisfies our hypotheses but does not satisfy assumptions introduced in previous works. A Brezis’s type nonexistence result is also given for general (smooth) elliptic-hyperbolic systems.Elsevier10000-01-01T00:00:00Z2018-09-15T00:00:00Z2018-09-15info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10773/23634eng0022-247X10.1016/j.jmaa.2018.05.047Leonetti, FrancescoRocha, EugenioStaicu, Vasileinfo: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:RCAAP2024-02-22T11:46:33Zoai:ria.ua.pt:10773/23634Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T02:57:35.194175Repositório 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 |
Smallness and cancellation in some elliptic systems with measure data |
title |
Smallness and cancellation in some elliptic systems with measure data |
spellingShingle |
Smallness and cancellation in some elliptic systems with measure data Leonetti, Francesco Elliptic System Existence Measure Solution |
title_short |
Smallness and cancellation in some elliptic systems with measure data |
title_full |
Smallness and cancellation in some elliptic systems with measure data |
title_fullStr |
Smallness and cancellation in some elliptic systems with measure data |
title_full_unstemmed |
Smallness and cancellation in some elliptic systems with measure data |
title_sort |
Smallness and cancellation in some elliptic systems with measure data |
author |
Leonetti, Francesco |
author_facet |
Leonetti, Francesco Rocha, Eugenio Staicu, Vasile |
author_role |
author |
author2 |
Rocha, Eugenio Staicu, Vasile |
author2_role |
author author |
dc.contributor.author.fl_str_mv |
Leonetti, Francesco Rocha, Eugenio Staicu, Vasile |
dc.subject.por.fl_str_mv |
Elliptic System Existence Measure Solution |
topic |
Elliptic System Existence Measure Solution |
description |
In a bounded open subset Ω ⊂ Rn, we study Dirichlet problems with elliptic systems, involving a finite Radon measure μ on Rn with values into RN , defined by { −div A(x, u(x), Du(x)) = μ in Ω, u = 0 on ∂Ω, where Aα i (x, y, ξ) = N∑ β=1 n∑ j=1 aα,β i,j (x, y) ξβ j with α ∈ {1, . . . , N } the equation index. We prove the existence of a (distributional) solution u : Ω → RN , obtained as the limit of approximations, by assuming: (i) that coefficients aα,β i,j are bounded Carathéodory functions; (ii) ellipticity of the diagonal coefficients aα,α i,j ; and (iii) smallness of the quadratic form associated to the off-diagonal coefficients aα,β i,j (i.e. α = β) verifying a r-staircase support condition with r > 0. Such a smallness condition is satisfied, for instance, in each one of these cases: (a) aα,β i,j = −aβ,α j,i (skew-symmetry); (b) |aα,β i,j | is small; (c) aα,β i,j may be decomposed into two parts, the first enjoying skew-symmetry and the second being small in absolute value. We give an example that satisfies our hypotheses but does not satisfy assumptions introduced in previous works. A Brezis’s type nonexistence result is also given for general (smooth) elliptic-hyperbolic systems. |
publishDate |
2018 |
dc.date.none.fl_str_mv |
10000-01-01T00:00:00Z 2018-09-15T00:00:00Z 2018-09-15 |
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/10773/23634 |
url |
http://hdl.handle.net/10773/23634 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
0022-247X 10.1016/j.jmaa.2018.05.047 |
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 |
instname_str |
Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
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 |
repository.mail.fl_str_mv |
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1799137630106943488 |