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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Autor principal:	Rossi, Julio Daniel
Publicado:	2008
Materias:	Neumann boundary conditions Nonlocal diffusion p-Laplacian Total variation flow
Acceso en línea:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00217824_v90_n2_p201_Andreu http://hdl.handle.net/20.500.12110/paper_00217824_v90_n2_p201_Andreu
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	paper:paper_00217824_v90_n2_p201_Andreu
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spelling	paper:paper_00217824_v90_n2_p201_Andreu2025-07-30T17:24:29Z A nonlocal p-Laplacian evolution equation with Neumann boundary conditions Rossi, Julio Daniel 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. 2008 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00217824_v90_n2_p201_Andreu 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 Rossi, Julio Daniel 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.
author	Rossi, Julio Daniel
author_facet	Rossi, Julio Daniel
author_sort	Rossi, Julio Daniel
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	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00217824_v90_n2_p201_Andreu http://hdl.handle.net/20.500.12110/paper_00217824_v90_n2_p201_Andreu
work_keys_str_mv	AT rossijuliodaniel anonlocalplaplacianevolutionequationwithneumannboundaryconditions AT rossijuliodaniel nonlocalplaplacianevolutionequationwithneumannboundaryconditions
_version_	1840326180375363584

A nonlocal p-Laplacian evolution equation with Neumann boundary conditions

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