Fractional problems in thin domains

In this paper we consider nonlocal fractional problems in thin domains. Given open bounded subsets U⊂R n and V⊂R m , we show that the solution u ε to Δ x s u ε (x,y)+Δ y t u ε (x,y)=f(x,ε −1 y)inU×εV with u ε (x,y)=0 if x⁄∈U and y∈εV, verifies that ũ ε (x,y)≔u ε (x,εy)→u 0 strongly in the natural fr...

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Publicado:	2019
Materias:	Dirichlet problem Neumann problem Nonlocal fractional equations Thin domains Boundary value problems Dirichlet condition Fractional equation Fractional Laplacian Open bounded subsets Rate of convergence Sobolev spaces
Acceso en línea:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0362546X_v_n_p_Pereira http://hdl.handle.net/20.500.12110/paper_0362546X_v_n_p_Pereira
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	paper:paper_0362546X_v_n_p_Pereira
record_format	dspace
spelling	paper:paper_0362546X_v_n_p_Pereira2023-06-08T15:35:24Z Fractional problems in thin domains Dirichlet problem Neumann problem Nonlocal fractional equations Thin domains Boundary value problems Dirichlet condition Dirichlet problem Fractional equation Fractional Laplacian Neumann problem Open bounded subsets Rate of convergence Thin domains Sobolev spaces In this paper we consider nonlocal fractional problems in thin domains. Given open bounded subsets U⊂R n and V⊂R m , we show that the solution u ε to Δ x s u ε (x,y)+Δ y t u ε (x,y)=f(x,ε −1 y)inU×εV with u ε (x,y)=0 if x⁄∈U and y∈εV, verifies that ũ ε (x,y)≔u ε (x,εy)→u 0 strongly in the natural fractional Sobolev space associated to this problem. We also identify the limit problem that is satisfied by u 0 and estimate the rate of convergence in the uniform norm. Here Δ x s u and Δ y t u are the fractional Laplacian in the 1st variable x (with a Dirichlet condition, u(x)=0 if x⁄∈U) and in the 2nd variable y (with a Neumann condition, integrating only inside V), respectively, that is, Δ x s u(x,y)=∫ R n [Formula presented]dw and Δ y t u(x,y)=∫ V [Formula presented]dz. © 2019 Elsevier Ltd 2019 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0362546X_v_n_p_Pereira http://hdl.handle.net/20.500.12110/paper_0362546X_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 fractional equations Thin domains Boundary value problems Dirichlet condition Dirichlet problem Fractional equation Fractional Laplacian Neumann problem Open bounded subsets Rate of convergence Thin domains Sobolev spaces
spellingShingle	Dirichlet problem Neumann problem Nonlocal fractional equations Thin domains Boundary value problems Dirichlet condition Dirichlet problem Fractional equation Fractional Laplacian Neumann problem Open bounded subsets Rate of convergence Thin domains Sobolev spaces Fractional problems in thin domains
topic_facet	Dirichlet problem Neumann problem Nonlocal fractional equations Thin domains Boundary value problems Dirichlet condition Dirichlet problem Fractional equation Fractional Laplacian Neumann problem Open bounded subsets Rate of convergence Thin domains Sobolev spaces
description	In this paper we consider nonlocal fractional problems in thin domains. Given open bounded subsets U⊂R n and V⊂R m , we show that the solution u ε to Δ x s u ε (x,y)+Δ y t u ε (x,y)=f(x,ε −1 y)inU×εV with u ε (x,y)=0 if x⁄∈U and y∈εV, verifies that ũ ε (x,y)≔u ε (x,εy)→u 0 strongly in the natural fractional Sobolev space associated to this problem. We also identify the limit problem that is satisfied by u 0 and estimate the rate of convergence in the uniform norm. Here Δ x s u and Δ y t u are the fractional Laplacian in the 1st variable x (with a Dirichlet condition, u(x)=0 if x⁄∈U) and in the 2nd variable y (with a Neumann condition, integrating only inside V), respectively, that is, Δ x s u(x,y)=∫ R n [Formula presented]dw and Δ y t u(x,y)=∫ V [Formula presented]dz. © 2019 Elsevier Ltd
title	Fractional problems in thin domains
title_short	Fractional problems in thin domains
title_full	Fractional problems in thin domains
title_fullStr	Fractional problems in thin domains
title_full_unstemmed	Fractional problems in thin domains
title_sort	fractional problems in thin domains
publishDate	2019
url	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0362546X_v_n_p_Pereira http://hdl.handle.net/20.500.12110/paper_0362546X_v_n_p_Pereira
_version_	1768546161899077632

Fractional problems in thin domains

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