Nonlocal Lagrange multipliers and transport densities
Autor(a) principal: | |
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Data de Publicação: | 2023 |
Outros Autores: | , |
Tipo de documento: | Artigo |
Idioma: | por |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | https://hdl.handle.net/1822/88311 |
Resumo: | We prove the existence of generalised solutions of the Monge-Kantorovich equations with fractional $s$-gradient constraint, $0<s<1$, associated to a general, possibly degenerate, linear fractional operator of the type, \begin{equation*} \mathscr L^su=-D^s\cdot(AD^su+\bs bu)+\bs d\cdot D^su+cu , \end{equation*} with integrable data, in the space $\Lambda^{s,p}_0(\Omega)$, which is the completion of the set of smooth functions with compact support in a bounded domain $\Omega$ for the $L^p$-norm of the distributional Riesz fractional gradient $D^s$ in $\R^d$ (when $s=1$, $D^1=D$ is the classical gradient). The transport densities arise as generalised Lagrange multipliers in the dual space of $L^\infty(\R^d)$ and are associated to the variational inequalities of the corresponding transport potentials under the constraint $|D^su|\leq g$. Their existence is shown by approximating the variational inequality through a penalisation of the constraint and nonlinear regularisation of the linear operator $\mathscr L^su$. For this purpose, we also develop some relevant properties of the spaces $\Lambda^{s,p}_0(\Omega)$, including the limit case $p=\infty$ and the continuous embeddings $\Lambda^{s,q}_0(\Omega)\subset \Lambda^{s,p}_0(\Omega)$, for $1\le p\le q\le\infty$. We also show the localisation of the nonlocal problems ($0<s<1$), to the local limit problem with classical gradient constraint when $s\rightarrow1$, for which most results are also new for a general, possibly degenerate, partial differential operator $\mathscr L^1u$ only with integrable coefficients and bounded gradient constraint. |
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Nonlocal Lagrange multipliers and transport densitiesFractional gradientNonlocal variational inequalitiesLagrange multipliersCiências Naturais::MatemáticasWe prove the existence of generalised solutions of the Monge-Kantorovich equations with fractional $s$-gradient constraint, $0<s<1$, associated to a general, possibly degenerate, linear fractional operator of the type, \begin{equation*} \mathscr L^su=-D^s\cdot(AD^su+\bs bu)+\bs d\cdot D^su+cu , \end{equation*} with integrable data, in the space $\Lambda^{s,p}_0(\Omega)$, which is the completion of the set of smooth functions with compact support in a bounded domain $\Omega$ for the $L^p$-norm of the distributional Riesz fractional gradient $D^s$ in $\R^d$ (when $s=1$, $D^1=D$ is the classical gradient). The transport densities arise as generalised Lagrange multipliers in the dual space of $L^\infty(\R^d)$ and are associated to the variational inequalities of the corresponding transport potentials under the constraint $|D^su|\leq g$. Their existence is shown by approximating the variational inequality through a penalisation of the constraint and nonlinear regularisation of the linear operator $\mathscr L^su$. For this purpose, we also develop some relevant properties of the spaces $\Lambda^{s,p}_0(\Omega)$, including the limit case $p=\infty$ and the continuous embeddings $\Lambda^{s,q}_0(\Omega)\subset \Lambda^{s,p}_0(\Omega)$, for $1\le p\le q\le\infty$. We also show the localisation of the nonlocal problems ($0<s<1$), to the local limit problem with classical gradient constraint when $s\rightarrow1$, for which most results are also new for a general, possibly degenerate, partial differential operator $\mathscr L^1u$ only with integrable coefficients and bounded gradient constraint.The research of J. F. Rodrigues was partially done under the framework of the Project PTDC/MAT-PUR/28686/2017 at CMAFcIO/ULisboa and A. Azevedo and L. Santos were par tially nanced by Portuguese Funds through FCT (Fundação para a Ciência e a Tecnologia) within the Projects UIDB/00013/2020 and UIDP/00013/2020.World ScientificUniversidade do MinhoAzevedo, AssisRodrigues, José FranciscoSantos, Lisa20232023-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/1822/88311porAzevedo, A., Rodrigues, J. F., & Santos, L. (2023, December 8). Nonlocal Lagrange multipliers and transport densities. Bulletin of Mathematical Sciences. World Scientific Pub Co Pte Ltd. http://doi.org/10.1142/s16643607235001451664-36071664-361510.1142/S1664360723500145https://www.worldscientific.com/doi/10.1142/S1664360723500145info: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:RCAAP2024-02-03T01:20:20Zoai:repositorium.sdum.uminho.pt:1822/88311Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T02:07:29.241227Repositório 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 |
Nonlocal Lagrange multipliers and transport densities |
title |
Nonlocal Lagrange multipliers and transport densities |
spellingShingle |
Nonlocal Lagrange multipliers and transport densities Azevedo, Assis Fractional gradient Nonlocal variational inequalities Lagrange multipliers Ciências Naturais::Matemáticas |
title_short |
Nonlocal Lagrange multipliers and transport densities |
title_full |
Nonlocal Lagrange multipliers and transport densities |
title_fullStr |
Nonlocal Lagrange multipliers and transport densities |
title_full_unstemmed |
Nonlocal Lagrange multipliers and transport densities |
title_sort |
Nonlocal Lagrange multipliers and transport densities |
author |
Azevedo, Assis |
author_facet |
Azevedo, Assis Rodrigues, José Francisco Santos, Lisa |
author_role |
author |
author2 |
Rodrigues, José Francisco Santos, Lisa |
author2_role |
author author |
dc.contributor.none.fl_str_mv |
Universidade do Minho |
dc.contributor.author.fl_str_mv |
Azevedo, Assis Rodrigues, José Francisco Santos, Lisa |
dc.subject.por.fl_str_mv |
Fractional gradient Nonlocal variational inequalities Lagrange multipliers Ciências Naturais::Matemáticas |
topic |
Fractional gradient Nonlocal variational inequalities Lagrange multipliers Ciências Naturais::Matemáticas |
description |
We prove the existence of generalised solutions of the Monge-Kantorovich equations with fractional $s$-gradient constraint, $0<s<1$, associated to a general, possibly degenerate, linear fractional operator of the type, \begin{equation*} \mathscr L^su=-D^s\cdot(AD^su+\bs bu)+\bs d\cdot D^su+cu , \end{equation*} with integrable data, in the space $\Lambda^{s,p}_0(\Omega)$, which is the completion of the set of smooth functions with compact support in a bounded domain $\Omega$ for the $L^p$-norm of the distributional Riesz fractional gradient $D^s$ in $\R^d$ (when $s=1$, $D^1=D$ is the classical gradient). The transport densities arise as generalised Lagrange multipliers in the dual space of $L^\infty(\R^d)$ and are associated to the variational inequalities of the corresponding transport potentials under the constraint $|D^su|\leq g$. Their existence is shown by approximating the variational inequality through a penalisation of the constraint and nonlinear regularisation of the linear operator $\mathscr L^su$. For this purpose, we also develop some relevant properties of the spaces $\Lambda^{s,p}_0(\Omega)$, including the limit case $p=\infty$ and the continuous embeddings $\Lambda^{s,q}_0(\Omega)\subset \Lambda^{s,p}_0(\Omega)$, for $1\le p\le q\le\infty$. We also show the localisation of the nonlocal problems ($0<s<1$), to the local limit problem with classical gradient constraint when $s\rightarrow1$, for which most results are also new for a general, possibly degenerate, partial differential operator $\mathscr L^1u$ only with integrable coefficients and bounded gradient constraint. |
publishDate |
2023 |
dc.date.none.fl_str_mv |
2023 2023-01-01T00:00:00Z |
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 |
https://hdl.handle.net/1822/88311 |
url |
https://hdl.handle.net/1822/88311 |
dc.language.iso.fl_str_mv |
por |
language |
por |
dc.relation.none.fl_str_mv |
Azevedo, A., Rodrigues, J. F., & Santos, L. (2023, December 8). Nonlocal Lagrange multipliers and transport densities. Bulletin of Mathematical Sciences. World Scientific Pub Co Pte Ltd. http://doi.org/10.1142/s1664360723500145 1664-3607 1664-3615 10.1142/S1664360723500145 https://www.worldscientific.com/doi/10.1142/S1664360723500145 |
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 |
World Scientific |
publisher.none.fl_str_mv |
World Scientific |
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 |
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 |
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1799137163350114304 |