Asymptotic behavior of the interface for entire vector minimizers in phase transitions

We study globally bounded entire minimizers $u:\mathbb{R}^n\rightarrow\mathbb{R}^m$ of Allen-Cahn systems for potentials $W\geq 0$ with $\{W=0\}=\{a_1,...,a_N\}$ and $W(u)\sim |u-a_i|^\alpha$ near $u=a_i$, $0<\alpha<2$. Such solutions are, over large regions, identically equal to some zeroes o...

Descripción completa

Detalles Bibliográficos
Autores: Alikakos, N., Geng, Z., Zarnescu, A.
Tipo de recurso: artículo
Estado:Versión aceptada para publicación
Fecha de publicación:2022
País:España
Institución:Basque Center for Applied Mathematics (BCAM)
Repositorio:BIRD. BCAM's Institutional Repository Data
OAI Identifier:oai:bird.bcamath.org:20.500.11824/1632
Acceso en línea:http://hdl.handle.net/20.500.11824/1632
Access Level:acceso abierto
Palabra clave:Vector minimizers, phase transitions, subquadratic potential, entire solutions.
id ES_b10cf5516ca3f204df0c17cf5e7acd64
oai_identifier_str oai:bird.bcamath.org:20.500.11824/1632
network_acronym_str ES
network_name_str España
repository_id_str
spelling Asymptotic behavior of the interface for entire vector minimizers in phase transitionsAlikakos, N.Geng, Z.Zarnescu, A.Vector minimizers, phase transitions, subquadratic potential, entire solutions.We study globally bounded entire minimizers $u:\mathbb{R}^n\rightarrow\mathbb{R}^m$ of Allen-Cahn systems for potentials $W\geq 0$ with $\{W=0\}=\{a_1,...,a_N\}$ and $W(u)\sim |u-a_i|^\alpha$ near $u=a_i$, $0<\alpha<2$. Such solutions are, over large regions, identically equal to some zeroes of the potential $a_i$'s. We establish the estimates $$ \mathcal{L}^n(I_0\cap B_r(x_0))\leq c_1r^{n-1},\quad \mathcal{H}^{n-1}(\partial^* I_0\cap B_r(x_0))\geq c_2r^{n-1}, \quad r\geq r_0(x_0) $$ for the diffuse interface $I_0:=\{x\in\mathbb{R}^n: \min_{1\leq i\leq N}|u(x)-a_i|>0\}$ and the free boundary $\partial I_0$. Furthermore, if $\alpha=1$ we establish the upper bound $$ \mathcal{H}^{n-1}(\partial^* I_0\cap B_r(x_0))\leq c_3r^{n-1}, \quad r\geq r_0(x_0). $$202320232022info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersionapplication/pdfhttp://hdl.handle.net/20.500.11824/1632reponame:BIRD. BCAM's Institutional Repository Datainstname:Basque Center for Applied Mathematics (BCAM)Ingléshttps://www.sciencedirect.com/science/article/abs/pii/S0022123622001859info:eu-repo/grantAgreement/MINECO//SEV-2017-0718info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/PID2020-114189RB-I00info:eu-repo/grantAgreement/Gobierno Vasco/BERC/BERC.2022-2025Reconocimiento-NoComercial-CompartirIgual 3.0 Españahttp://creativecommons.org/licenses/by-nc-sa/3.0/es/info:eu-repo/semantics/openAccessoai:bird.bcamath.org:20.500.11824/16322026-06-19T12:47:47Z
dc.title.none.fl_str_mv Asymptotic behavior of the interface for entire vector minimizers in phase transitions
title Asymptotic behavior of the interface for entire vector minimizers in phase transitions
spellingShingle Asymptotic behavior of the interface for entire vector minimizers in phase transitions
Alikakos, N.
Vector minimizers, phase transitions, subquadratic potential, entire solutions.
title_short Asymptotic behavior of the interface for entire vector minimizers in phase transitions
title_full Asymptotic behavior of the interface for entire vector minimizers in phase transitions
title_fullStr Asymptotic behavior of the interface for entire vector minimizers in phase transitions
title_full_unstemmed Asymptotic behavior of the interface for entire vector minimizers in phase transitions
title_sort Asymptotic behavior of the interface for entire vector minimizers in phase transitions
dc.creator.none.fl_str_mv Alikakos, N.
Geng, Z.
Zarnescu, A.
author Alikakos, N.
author_facet Alikakos, N.
Geng, Z.
Zarnescu, A.
author_role author
author2 Geng, Z.
Zarnescu, A.
author2_role author
author
dc.subject.none.fl_str_mv Vector minimizers, phase transitions, subquadratic potential, entire solutions.
topic Vector minimizers, phase transitions, subquadratic potential, entire solutions.
description We study globally bounded entire minimizers $u:\mathbb{R}^n\rightarrow\mathbb{R}^m$ of Allen-Cahn systems for potentials $W\geq 0$ with $\{W=0\}=\{a_1,...,a_N\}$ and $W(u)\sim |u-a_i|^\alpha$ near $u=a_i$, $0<\alpha<2$. Such solutions are, over large regions, identically equal to some zeroes of the potential $a_i$'s. We establish the estimates $$ \mathcal{L}^n(I_0\cap B_r(x_0))\leq c_1r^{n-1},\quad \mathcal{H}^{n-1}(\partial^* I_0\cap B_r(x_0))\geq c_2r^{n-1}, \quad r\geq r_0(x_0) $$ for the diffuse interface $I_0:=\{x\in\mathbb{R}^n: \min_{1\leq i\leq N}|u(x)-a_i|>0\}$ and the free boundary $\partial I_0$. Furthermore, if $\alpha=1$ we establish the upper bound $$ \mathcal{H}^{n-1}(\partial^* I_0\cap B_r(x_0))\leq c_3r^{n-1}, \quad r\geq r_0(x_0). $$
publishDate 2022
dc.date.none.fl_str_mv 2022
2023
2023
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/acceptedVersion
format article
status_str acceptedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/20.500.11824/1632
url http://hdl.handle.net/20.500.11824/1632
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv https://www.sciencedirect.com/science/article/abs/pii/S0022123622001859
info:eu-repo/grantAgreement/MINECO//SEV-2017-0718
info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/PID2020-114189RB-I00
info:eu-repo/grantAgreement/Gobierno Vasco/BERC/BERC.2022-2025
dc.rights.none.fl_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:BIRD. BCAM's Institutional Repository Data
instname:Basque Center for Applied Mathematics (BCAM)
instname_str Basque Center for Applied Mathematics (BCAM)
reponame_str BIRD. BCAM's Institutional Repository Data
collection BIRD. BCAM's Institutional Repository Data
repository.name.fl_str_mv
repository.mail.fl_str_mv
_version_ 1869416878234927104
score 15.198674

Similares en LA Referencia