A nonlocal p-Laplacian evolution equation with Neumann boundary conditions

LOCAL AND NONLOCAL WEIGHTED p-LAPLACIAN EVOLUTION EQUATIONS WITH NEUMANN BOUNDARY CONDITIONS

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)...

Nonlocal Problems with Neumann Boundary Conditions

We introduce a new Neumann problem for the fractional Laplacian arising from a simple probabilistic consideration, and we discuss the basic properties of this model. We can consider both elliptic and parabolic equations in any domain. In addition, we formulate problems with nonhomogeneous Neumann conditions, and also with mixed Dirichlet and Neumann conditions, all of them having a clear probab...

A FREE-BOUNDARY PROBLEM FOR THE EVOLUTION p-LAPLACIAN EQUATION WITH A COMBUSTION BOUNDARY CONDITION

We study the existence, uniqueness and regularity of solutions of the equation ft = ∆pf = div (|Df | p−2 Df) under over-determined boundary conditions f = 0 and |Df | = 1. We show that if the initial data is concave and Lipschitz with a bounded and convex support, then the problem admits a unique solution which exists until it vanishes identically. Furthermore, the free-boundary of the support ...

How to Approximate the Heat Equation with Neumann Boundary Conditions by Nonlocal Diffusion Problems

We present a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary. We study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero. We prove that the solutions of this family of problems converge to a solution of the heat equation...