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

This is the authors' abstract. We don't add key points for this paper.

In this paper we study the nonlocal p-Laplacian-type diffusion equation uₜ(t,x)=∫RNJ(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[×(RNΩ). 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.