A NONLOCAL p-LAPLACIAN EVOLUTION EQUATION WITH NONHOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS

In this paper we study the nonlocal $p$-Laplacian-type diffusion equation $u_t(t,x)=\int_{\mathbb{R}^N}J(x-y)|u(t,y)-u(t,x)|^{p-2}(u(t,y)-u(t,x))\,dy$, $(t,x)\in]0,T[\times\Omega$, with $u(t,x)=\psi(x)$ for $(t,x)\in]0,T[\times(\mathbb{R}^N\setminus\Omega)$. If $p>1$, this is the nonlocal analogous problem to the well-known local $p$-Laplacian evolution equation $u_t=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ with Dirichlet boundary condition $u(t,x)=\psi(x)$ on $(t,x)\in]0,T[\times\partial\Omega$. If $p=1$, this is the nonlocal analogous to the total variation flow. When $p=+\infty$ (this has to be interpreted as the limit as $p\to+\infty$ 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 $\Omega$. 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.