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...

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Detalles Bibliográficos
Publicado: 2019
Materias:
Dirichlet problem
Neumann problem
nonlocal equations
perforated domains
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03082105_v_n_p_Pereira
http://hdl.handle.net/20.500.12110/paper_03082105_v_n_p_Pereira
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_03082105_v_n_p_Pereira
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spelling paper:paper_03082105_v_n_p_Pereira2023-06-08T15:31:37Z Nonlocal problems in perforated domains 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. 2019 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03082105_v_n_p_Pereira 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
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.
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
publishDate 2019
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03082105_v_n_p_Pereira
http://hdl.handle.net/20.500.12110/paper_03082105_v_n_p_Pereira
_version_ 1768545189273534464

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