Abstract:
In this paper we study the nonlocal p-Laplacian type diffusion equation,ut (t, x) = under(∫, Ω) J (x - y) | u (t, y) - u (t, x) |p - 2 (u (t, y) - u (t, x)) d y . If p > 1, this is the nonlocal analogous problem to the well-known local p-Laplacian evolution equation ut = div (| ∇ u |p - 2 ∇ u) with homogeneous Neumann boundary conditions. We prove existence and uniqueness of a strong solution, and if the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L∞ (0, T ; Lp (Ω)) to the solution of the p-Laplacian with homogeneous Neumann boundary conditions. The extreme case p = 1, that is, the nonlocal analogous to the total variation flow, is also analyzed. Finally, we study the asymptotic behavior of the solutions as t goes to infinity, showing the convergence to the mean value of the initial condition. © 2008 Elsevier Masson SAS. All rights reserved.
Registro:
Documento: |
Artículo
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Título: | A nonlocal p-Laplacian evolution equation with Neumann boundary conditions |
Autor: | Andreu, F.; Mazón, J.M.; Rossi, J.D.; Toledo, J. |
Filiación: | Departament de Matemàtica Aplicada, Universitat de València, Valencia, Spain Departament d'Anàlisi Matemàtica, Universitat de València, Valencia, Spain IMDEA Matematicas, Universitat Autonoma, C-IX, Madrid, Spain Departamento de Matemática, FCEyN UBA, 1428 Buenos Aires, Argentina
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Palabras clave: | Neumann boundary conditions; Nonlocal diffusion; p-Laplacian; Total variation flow |
Año: | 2008
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Volumen: | 90
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Número: | 2
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Página de inicio: | 201
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Página de fin: | 227
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DOI: |
http://dx.doi.org/10.1016/j.matpur.2008.04.003 |
Título revista: | Journal des Mathematiques Pures et Appliquees
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Título revista abreviado: | J. Math. Pures Appl.
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ISSN: | 00217824
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CODEN: | JMPAA
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PDF: | https://bibliotecadigital.exactas.uba.ar/download/paper/paper_00217824_v90_n2_p201_Andreu.pdf |
Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00217824_v90_n2_p201_Andreu |
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Citas:
---------- APA ----------
Andreu, F., Mazón, J.M., Rossi, J.D. & Toledo, J.
(2008)
. A nonlocal p-Laplacian evolution equation with Neumann boundary conditions. Journal des Mathematiques Pures et Appliquees, 90(2), 201-227.
http://dx.doi.org/10.1016/j.matpur.2008.04.003---------- CHICAGO ----------
Andreu, F., Mazón, J.M., Rossi, J.D., Toledo, J.
"A nonlocal p-Laplacian evolution equation with Neumann boundary conditions"
. Journal des Mathematiques Pures et Appliquees 90, no. 2
(2008) : 201-227.
http://dx.doi.org/10.1016/j.matpur.2008.04.003---------- MLA ----------
Andreu, F., Mazón, J.M., Rossi, J.D., Toledo, J.
"A nonlocal p-Laplacian evolution equation with Neumann boundary conditions"
. Journal des Mathematiques Pures et Appliquees, vol. 90, no. 2, 2008, pp. 201-227.
http://dx.doi.org/10.1016/j.matpur.2008.04.003---------- VANCOUVER ----------
Andreu, F., Mazón, J.M., Rossi, J.D., Toledo, J. A nonlocal p-Laplacian evolution equation with Neumann boundary conditions. J. Math. Pures Appl. 2008;90(2):201-227.
http://dx.doi.org/10.1016/j.matpur.2008.04.003