Nonlocal diffusion equations in Carnot Groups

Abstract

Let $G$ be a Carnot group. We study nonlocal diffusion equations in a domain $\Omega$ of $G$ of the form $$ u_t^\epsilon(x,t)=\int_{G}\frac{1}{\epsilon^2}K_{\epsilon}(x,y)(u^\epsilon(y,t)-u^\epsilon(x,t))\,dy, \qquad x\in \Omega $$ with $u^\epsilon=g(x,t)$ for $x\notin\Omega$. For appropriate rescaled kernel $K_\epsilon$ we prove that solutions $u^\epsilon$, when $\epsilon\rightarrow0$, uniformly approximate the solution of different local Dirichlet problem in $G$. The key tool used is the Taylor series development for a function defined on a Carnot group.