A nonlocal p-laplacian evolution equation with nonhomogeneous dirichlet boundary conditions
In this paper we study the nonlocal p-Laplacian- type diffusion equation ut(t,x) = ∫RN J(x-y)|u(t, y) -u(t,x)| p-2(u(t, y) -u(t,x)) dy, (t, x) ∈]0,T[×ω, with u(t, x) = ψ(x) for (t, x) ∈ ]0,T[×(RN\\ω). If p > 1, this is the nonlocal analogous problem to the well-known local p- Laplacian evolut...
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todo:paper_00361410_v40_n5_p1815_Andreu2023-10-03T14:47:36Z A nonlocal p-laplacian evolution equation with nonhomogeneous dirichlet boundary conditions Andreu, F. Mazón, J.M. Rossi, J.D. Toledo, J. Nonhomogeneous Dirichlet boundary conditions Nonlocal diffusion P-Laplacian Sandpiles Total variation flow Nonhomogeneous Dirichlet boundary conditions Nonlocal diffusion P-Laplacian Sandpiles Total variation flow Boundary conditions Diffusion Laplace transforms Sand In this paper we study the nonlocal p-Laplacian- type diffusion equation ut(t,x) = ∫RN J(x-y)|u(t, y) -u(t,x)| p-2(u(t, y) -u(t,x)) dy, (t, x) ∈]0,T[×ω, with u(t, x) = ψ(x) for (t, x) ∈ ]0,T[×(RN\\ω). If p > 1, this is the nonlocal analogous problem to the well-known local p- Laplacian evolution equation ut = div(| δu|p-2 δu) with Dirichlet boundary condition u(t, x) =ψ(x) on (t, x) ∈ ]0,T[×∂ω. If p = 1, this is the nonlocal analogous to the total variation flow. When p = +∞ (this has to be interpreted as the limit as p → +∞ in the previous model) we find an evolution problem that can be seen as a nonlocal model for the formation of sandpiles (here u(t,x) stands for the height of the sandpile) with prescribed height of sand outside of ω. We prove, as main results, existence, uniqueness, a contraction property that gives well posedness of the problem, and the convergence of the solutions to solutions of the local analogous problem when a rescaling parameter goes to zero. © 2009 Society for Industrial and Applied Mathematics. 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_00361410_v40_n5_p1815_Andreu |
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Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Nonhomogeneous Dirichlet boundary conditions Nonlocal diffusion P-Laplacian Sandpiles Total variation flow Nonhomogeneous Dirichlet boundary conditions Nonlocal diffusion P-Laplacian Sandpiles Total variation flow Boundary conditions Diffusion Laplace transforms Sand |
| spellingShingle |
Nonhomogeneous Dirichlet boundary conditions Nonlocal diffusion P-Laplacian Sandpiles Total variation flow Nonhomogeneous Dirichlet boundary conditions Nonlocal diffusion P-Laplacian Sandpiles Total variation flow Boundary conditions Diffusion Laplace transforms Sand Andreu, F. Mazón, J.M. Rossi, J.D. Toledo, J. A nonlocal p-laplacian evolution equation with nonhomogeneous dirichlet boundary conditions |
| topic_facet |
Nonhomogeneous Dirichlet boundary conditions Nonlocal diffusion P-Laplacian Sandpiles Total variation flow Nonhomogeneous Dirichlet boundary conditions Nonlocal diffusion P-Laplacian Sandpiles Total variation flow Boundary conditions Diffusion Laplace transforms Sand |
| description |
In this paper we study the nonlocal p-Laplacian- type diffusion equation ut(t,x) = ∫RN J(x-y)|u(t, y) -u(t,x)| p-2(u(t, y) -u(t,x)) dy, (t, x) ∈]0,T[×ω, with u(t, x) = ψ(x) for (t, x) ∈ ]0,T[×(RN\\ω). If p > 1, this is the nonlocal analogous problem to the well-known local p- Laplacian evolution equation ut = div(| δu|p-2 δu) with Dirichlet boundary condition u(t, x) =ψ(x) on (t, x) ∈ ]0,T[×∂ω. If p = 1, this is the nonlocal analogous to the total variation flow. When p = +∞ (this has to be interpreted as the limit as p → +∞ in the previous model) we find an evolution problem that can be seen as a nonlocal model for the formation of sandpiles (here u(t,x) stands for the height of the sandpile) with prescribed height of sand outside of ω. We prove, as main results, existence, uniqueness, a contraction property that gives well posedness of the problem, and the convergence of the solutions to solutions of the local analogous problem when a rescaling parameter goes to zero. © 2009 Society for Industrial and Applied Mathematics. |
| 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 nonhomogeneous dirichlet boundary conditions |
| title_short |
A nonlocal p-laplacian evolution equation with nonhomogeneous dirichlet boundary conditions |
| title_full |
A nonlocal p-laplacian evolution equation with nonhomogeneous dirichlet boundary conditions |
| title_fullStr |
A nonlocal p-laplacian evolution equation with nonhomogeneous dirichlet boundary conditions |
| title_full_unstemmed |
A nonlocal p-laplacian evolution equation with nonhomogeneous dirichlet boundary conditions |
| title_sort |
nonlocal p-laplacian evolution equation with nonhomogeneous dirichlet boundary conditions |
| url |
http://hdl.handle.net/20.500.12110/paper_00361410_v40_n5_p1815_Andreu |
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