A nonlocal p-Laplacian evolution equation with Neumann boundary conditions
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...
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2008
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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_00217824_v90_n2_p201_Andreu http://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_00217824_v90_n2_p201_Andreu_oai |
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I28-R145-paper_00217824_v90_n2_p201_Andreu_oai2020-10-19 Andreu, F. Mazón, J.M. Rossi, J.D. Toledo, J. 2008 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. Fil:Rossi, J.D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. application/pdf http://hdl.handle.net/20.500.12110/paper_00217824_v90_n2_p201_Andreu info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar J. Math. Pures Appl. 2008;90(2):201-227 Neumann boundary conditions Nonlocal diffusion p-Laplacian Total variation flow A nonlocal p-Laplacian evolution equation with Neumann boundary conditions info:eu-repo/semantics/article info:ar-repo/semantics/artículo info:eu-repo/semantics/publishedVersion http://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_00217824_v90_n2_p201_Andreu_oai |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-145 |
| collection |
Repositorio Digital de la Universidad de Buenos Aires (UBA) |
| topic |
Neumann boundary conditions Nonlocal diffusion p-Laplacian Total variation flow |
| spellingShingle |
Neumann boundary conditions Nonlocal diffusion p-Laplacian Total variation flow Andreu, F. Mazón, J.M. Rossi, J.D. Toledo, J. A nonlocal p-Laplacian evolution equation with Neumann boundary conditions |
| topic_facet |
Neumann boundary conditions Nonlocal diffusion p-Laplacian Total variation flow |
| description |
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. |
| format |
Artículo Artículo publishedVersion |
| author |
Andreu, F. Mazón, J.M. Rossi, J.D. Toledo, J. |
| author_facet |
Andreu, F. Mazón, J.M. Rossi, J.D. Toledo, J. |
| author_sort |
Andreu, F. |
| title |
A nonlocal p-Laplacian evolution equation with Neumann boundary conditions |
| title_short |
A nonlocal p-Laplacian evolution equation with Neumann boundary conditions |
| title_full |
A nonlocal p-Laplacian evolution equation with Neumann boundary conditions |
| title_fullStr |
A nonlocal p-Laplacian evolution equation with Neumann boundary conditions |
| title_full_unstemmed |
A nonlocal p-Laplacian evolution equation with Neumann boundary conditions |
| title_sort |
nonlocal p-laplacian evolution equation with neumann boundary conditions |
| publishDate |
2008 |
| url |
http://hdl.handle.net/20.500.12110/paper_00217824_v90_n2_p201_Andreu http://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_00217824_v90_n2_p201_Andreu_oai |
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