Nonlocal problems in thin domains

In this paper we consider nonlocal problems in thin domains. First, we deal with a nonlocal Neumann problem, that is, we study the behavior of the solutions to f(x)=∫Ω1×Ω2Jϵ(x−y)(uϵ(y)−uϵ(x))dy with Jϵ(z)=J(z1,ϵz2) and Ω=Ω1×Ω2⊂RN=RN1×RN2 a bounded domain. We find that there is a limit problem, that...

Descripción completa

Guardado en:
Detalles Bibliográficos
Autores principales: Pereira, M.C., Rossi, J.D.
Formato: JOUR
Materias:
Dirichlet problem
Neumann problem
Nonlocal equations
Thin domains
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_00220396_v263_n3_p1725_Pereira
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id todo:paper_00220396_v263_n3_p1725_Pereira
record_format dspace
spelling todo:paper_00220396_v263_n3_p1725_Pereira2023-10-03T14:25:37Z Nonlocal problems in thin domains Pereira, M.C. Rossi, J.D. Dirichlet problem Neumann problem Nonlocal equations Thin domains In this paper we consider nonlocal problems in thin domains. First, we deal with a nonlocal Neumann problem, that is, we study the behavior of the solutions to f(x)=∫Ω1×Ω2Jϵ(x−y)(uϵ(y)−uϵ(x))dy with Jϵ(z)=J(z1,ϵz2) and Ω=Ω1×Ω2⊂RN=RN1×RN2 a bounded domain. We find that there is a limit problem, that is, we show that uϵ→u0 as ϵ→0 in Ω and this limit function verifies ∫Ω2f(x1,x2)dx2=|Ω2|∫Ω1J(x1−y1,0)(U0(y1)−U0(x1))dy1, with U0(x1)=∫Ω2u0(x1,x2)dx2. In addition, we deal with a double limit when we add to this model a rescale in the kernel with a parameter that controls the size of the support of J. We show that this double limit exhibits some interesting features. We also study a nonlocal Dirichlet problem f(x)=∫RNJϵ(x−y)(uϵ(y)−uϵ(x))dy, x∈Ω, with uϵ(x)≡0, x∈RN∖Ω, and deal with similar issues. In this case the limit as ϵ→0 is u0=0 and the double limit problem commutes and also gives v≡0 at the end. © 2017 Elsevier Inc. Fil:Rossi, J.D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00220396_v263_n3_p1725_Pereira
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Dirichlet problem
Neumann problem
Nonlocal equations
Thin domains
spellingShingle Dirichlet problem
Neumann problem
Nonlocal equations
Thin domains
Pereira, M.C.
Rossi, J.D.
Nonlocal problems in thin domains
topic_facet Dirichlet problem
Neumann problem
Nonlocal equations
Thin domains
description In this paper we consider nonlocal problems in thin domains. First, we deal with a nonlocal Neumann problem, that is, we study the behavior of the solutions to f(x)=∫Ω1×Ω2Jϵ(x−y)(uϵ(y)−uϵ(x))dy with Jϵ(z)=J(z1,ϵz2) and Ω=Ω1×Ω2⊂RN=RN1×RN2 a bounded domain. We find that there is a limit problem, that is, we show that uϵ→u0 as ϵ→0 in Ω and this limit function verifies ∫Ω2f(x1,x2)dx2=|Ω2|∫Ω1J(x1−y1,0)(U0(y1)−U0(x1))dy1, with U0(x1)=∫Ω2u0(x1,x2)dx2. In addition, we deal with a double limit when we add to this model a rescale in the kernel with a parameter that controls the size of the support of J. We show that this double limit exhibits some interesting features. We also study a nonlocal Dirichlet problem f(x)=∫RNJϵ(x−y)(uϵ(y)−uϵ(x))dy, x∈Ω, with uϵ(x)≡0, x∈RN∖Ω, and deal with similar issues. In this case the limit as ϵ→0 is u0=0 and the double limit problem commutes and also gives v≡0 at the end. © 2017 Elsevier Inc.
format JOUR
author Pereira, M.C.
Rossi, J.D.
author_facet Pereira, M.C.
Rossi, J.D.
author_sort Pereira, M.C.
title Nonlocal problems in thin domains
title_short Nonlocal problems in thin domains
title_full Nonlocal problems in thin domains
title_fullStr Nonlocal problems in thin domains
title_full_unstemmed Nonlocal problems in thin domains
title_sort nonlocal problems in thin domains
url http://hdl.handle.net/20.500.12110/paper_00220396_v263_n3_p1725_Pereira
work_keys_str_mv AT pereiramc nonlocalproblemsinthindomains
AT rossijd nonlocalproblemsinthindomains
_version_ 1782027077541691392

Ejemplares similares