A nonlocal p-Laplacian evolution equation with Neumann boundary conditions

• Autores: Andreu, F.; Mazón, J.M.; Rossi, J.D.; Toledo, J.
• Año de publicación: 2008
• Idioma: inglés
• Tipo de recurso: artículo
• Fuente: J. Math. Pures Appl. 2008;90(2):201-227
• Materia: Neumann boundary conditions; Nonlocal diffusion; p-Laplacian; Total variation flow
• Institución: Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales

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.