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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todo:paper_00217824_v90_n2_p201_Andreu2023-10-03T14:20:51Z A nonlocal p-Laplacian evolution equation with Neumann boundary conditions Andreu, F. Mazón, J.M. Rossi, J.D. Toledo, J. Neumann boundary conditions Nonlocal diffusion p-Laplacian Total variation flow 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00217824_v90_n2_p201_Andreu |
institution |
Universidad de Buenos Aires |
institution_str |
I-28 |
repository_str |
R-134 |
collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (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 |
JOUR |
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 |
url |
http://hdl.handle.net/20.500.12110/paper_00217824_v90_n2_p201_Andreu |
work_keys_str_mv |
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