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Abstract:

In this paper, we study the nonlocal ∞-Laplacian type diffusion equation obtained as the limit as p → ∞ to the nonlocal analogous to the p-Laplacian evolution, u-t (t,x) = \\int-{\\mathbb{R}^N} J(x-y)|u(t,y) - u(t,x)|^{p-2}(u(t,y)- u(t,x)) \\, dy. We prove exist ence and uniqueness of a limit solution that verifies an equation governed by the subdifferential of a convex energy functional associated to the indicator function of the set {K = \\{ u \\in L2(\\mathbb{R}^N) \\, : \\, | u(x) - u(y)| \\le 1, \\mbox{ when } x-y \\in {\\rm supp} (J)\\}}. We also find some explicit examples of solutions to the limit equation. 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; L 2 (Ω)) to the limit solution of the local evolutions of the p-Laplacian, v t = Δ p v. This last limit problem has been proposed as a model to describe the formation of a sandpile. Moreover, we also analyze the collapse of the initial condition when it does not belong to K by means of a suitable rescale of the solution that describes the initial layer that appears for p large. Finally, we give an interpretation of the limit problem in terms of Monge-Kantorovich mass transport theory. © 2008 Springer-Verlag.

Registro:

Documento: Artículo
Título:The limit as p → ∞ in a nonlocal p-Laplacian evolution equation: A nonlocal approximation of a model for sandpiles
Autor:Andreu, F.; Mazón, J.M.; Rossi, J.D.; Toledo, J.
Filiación:Departament de Matemàtica Applicada, Universitat de València, Valencia, Spain
Departament d'Anàlisi Matemàtica, Universitat de València, Valencia, Spain
Departamento de Matemática, FCEyN UBA, Buenos Aires, Argentina
Año:2009
Volumen:35
Número:3
Página de inicio:279
Página de fin:316
DOI: http://dx.doi.org/10.1007/s00526-008-0205-2
Título revista:Calculus of Variations and Partial Differential Equations
Título revista abreviado:Calc. Var. Partial Differ. Equ.
ISSN:09442669
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_09442669_v35_n3_p279_Andreu

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Citas:

---------- APA ----------
Andreu, F., Mazón, J.M., Rossi, J.D. & Toledo, J. (2009) . The limit as p → ∞ in a nonlocal p-Laplacian evolution equation: A nonlocal approximation of a model for sandpiles. Calculus of Variations and Partial Differential Equations, 35(3), 279-316.
http://dx.doi.org/10.1007/s00526-008-0205-2
---------- CHICAGO ----------
Andreu, F., Mazón, J.M., Rossi, J.D., Toledo, J. "The limit as p → ∞ in a nonlocal p-Laplacian evolution equation: A nonlocal approximation of a model for sandpiles" . Calculus of Variations and Partial Differential Equations 35, no. 3 (2009) : 279-316.
http://dx.doi.org/10.1007/s00526-008-0205-2
---------- MLA ----------
Andreu, F., Mazón, J.M., Rossi, J.D., Toledo, J. "The limit as p → ∞ in a nonlocal p-Laplacian evolution equation: A nonlocal approximation of a model for sandpiles" . Calculus of Variations and Partial Differential Equations, vol. 35, no. 3, 2009, pp. 279-316.
http://dx.doi.org/10.1007/s00526-008-0205-2
---------- VANCOUVER ----------
Andreu, F., Mazón, J.M., Rossi, J.D., Toledo, J. The limit as p → ∞ in a nonlocal p-Laplacian evolution equation: A nonlocal approximation of a model for sandpiles. Calc. Var. Partial Differ. Equ. 2009;35(3):279-316.
http://dx.doi.org/10.1007/s00526-008-0205-2