A Nonlocal p-Laplacian Evolution Equation with Nonhomogeneous Dirichlet Boundary Conditions

Abstract. 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(|∇u|p−2∇u) 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.