Nonlocal problems in perforated domains

In this paper, we analyse nonlocal equations in perforated domains. We consider nonlocal problems of the form with x in a perforated domain. Here J is a nonsingular kernel. We think about as a fixed set ω from where we have removed a subset that we call the holes. We deal both with the Neumann and D...

Descripción completa

Guardado en:
Detalles Bibliográficos
Autores principales: Pereira, M.C., Rossi, J.D.
Formato: INPR
Materias:
Dirichlet problem
Neumann problem
nonlocal equations
perforated domains
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_03082105_v_n_p_Pereira
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id todo:paper_03082105_v_n_p_Pereira
record_format dspace
spelling todo:paper_03082105_v_n_p_Pereira2023-10-03T15:22:48Z Nonlocal problems in perforated domains Pereira, M.C. Rossi, J.D. Dirichlet problem Neumann problem nonlocal equations perforated domains In this paper, we analyse nonlocal equations in perforated domains. We consider nonlocal problems of the form with x in a perforated domain. Here J is a nonsingular kernel. We think about as a fixed set ω from where we have removed a subset that we call the holes. We deal both with the Neumann and Dirichlet conditions in the holes and assume a Dirichlet condition outside ω. In the latter case we impose that u vanishes in the holes but integrate in the whole ℝN (B = ℝN) and in the former we just consider integrals in ℝN minus the holes (B = ℝN \\ ω\\ωϵ). Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of has a weak limit, weakly∗ in L∞(ω), we analyse the limit as ϵ → 0 of the solutions to the nonlocal problems proving that there is a nonlocal limit problem. In the case in which the holes are periodically removed balls, we obtain that the critical radius is of the order of the size of the typical cell (that gives the period). In addition, in this periodic case, we also study the behaviour of these nonlocal problems when we rescale the kernel in order to approximate local PDE problems. © Royal Society of Edinburgh 2019. INPR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_03082105_v_n_p_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
perforated domains
spellingShingle Dirichlet problem
Neumann problem
nonlocal equations
perforated domains
Pereira, M.C.
Rossi, J.D.
Nonlocal problems in perforated domains
topic_facet Dirichlet problem
Neumann problem
nonlocal equations
perforated domains
description In this paper, we analyse nonlocal equations in perforated domains. We consider nonlocal problems of the form with x in a perforated domain. Here J is a nonsingular kernel. We think about as a fixed set ω from where we have removed a subset that we call the holes. We deal both with the Neumann and Dirichlet conditions in the holes and assume a Dirichlet condition outside ω. In the latter case we impose that u vanishes in the holes but integrate in the whole ℝN (B = ℝN) and in the former we just consider integrals in ℝN minus the holes (B = ℝN \\ ω\\ωϵ). Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of has a weak limit, weakly∗ in L∞(ω), we analyse the limit as ϵ → 0 of the solutions to the nonlocal problems proving that there is a nonlocal limit problem. In the case in which the holes are periodically removed balls, we obtain that the critical radius is of the order of the size of the typical cell (that gives the period). In addition, in this periodic case, we also study the behaviour of these nonlocal problems when we rescale the kernel in order to approximate local PDE problems. © Royal Society of Edinburgh 2019.
format INPR
author Pereira, M.C.
Rossi, J.D.
author_facet Pereira, M.C.
Rossi, J.D.
author_sort Pereira, M.C.
title Nonlocal problems in perforated domains
title_short Nonlocal problems in perforated domains
title_full Nonlocal problems in perforated domains
title_fullStr Nonlocal problems in perforated domains
title_full_unstemmed Nonlocal problems in perforated domains
title_sort nonlocal problems in perforated domains
url http://hdl.handle.net/20.500.12110/paper_03082105_v_n_p_Pereira
work_keys_str_mv AT pereiramc nonlocalproblemsinperforateddomains
AT rossijd nonlocalproblemsinperforateddomains
_version_ 1782028095452086272

Ejemplares similares