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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todo:paper_0362546X_v_n_p_Pereira2023-10-03T15:27:24Z Fractional problems in thin domains Pereira, M.C. Rossi, J.D. Saintier, N. 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 JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar 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 Pereira, M.C. Rossi, J.D. Saintier, N. 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 |
| format |
JOUR |
| author |
Pereira, M.C. Rossi, J.D. Saintier, N. |
| author_facet |
Pereira, M.C. Rossi, J.D. Saintier, N. |
| author_sort |
Pereira, M.C. |
| 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 |
| url |
http://hdl.handle.net/20.500.12110/paper_0362546X_v_n_p_Pereira |
| work_keys_str_mv |
AT pereiramc fractionalproblemsinthindomains AT rossijd fractionalproblemsinthindomains AT saintiern fractionalproblemsinthindomains |
| _version_ |
1807320066442657792 |