Abstract:
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.
Registro:
Documento: |
Artículo
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Título: | Nonlocal problems in thin domains |
Autor: | Pereira, M.C.; Rossi, J.D. |
Filiación: | Dpto. de Matemática Aplicada, IME, Universidade de São Paulo, Rua do Matão 1010, São Paulo, SP, Brazil Dpto. de Matemáticas, FCEyN, Universidad de Buenos Aires, Buenos Aires, 1428, Argentina
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Palabras clave: | Dirichlet problem; Neumann problem; Nonlocal equations; Thin domains |
Año: | 2017
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Volumen: | 263
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Número: | 3
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Página de inicio: | 1725
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Página de fin: | 1754
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DOI: |
http://dx.doi.org/10.1016/j.jde.2017.03.029 |
Título revista: | Journal of Differential Equations
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Título revista abreviado: | J. Differ. Equ.
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ISSN: | 00220396
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CODEN: | JDEQA
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Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00220396_v263_n3_p1725_Pereira |
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Citas:
---------- APA ----------
Pereira, M.C. & Rossi, J.D.
(2017)
. Nonlocal problems in thin domains. Journal of Differential Equations, 263(3), 1725-1754.
http://dx.doi.org/10.1016/j.jde.2017.03.029---------- CHICAGO ----------
Pereira, M.C., Rossi, J.D.
"Nonlocal problems in thin domains"
. Journal of Differential Equations 263, no. 3
(2017) : 1725-1754.
http://dx.doi.org/10.1016/j.jde.2017.03.029---------- MLA ----------
Pereira, M.C., Rossi, J.D.
"Nonlocal problems in thin domains"
. Journal of Differential Equations, vol. 263, no. 3, 2017, pp. 1725-1754.
http://dx.doi.org/10.1016/j.jde.2017.03.029---------- VANCOUVER ----------
Pereira, M.C., Rossi, J.D. Nonlocal problems in thin domains. J. Differ. Equ. 2017;263(3):1725-1754.
http://dx.doi.org/10.1016/j.jde.2017.03.029