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 $$\begin{aligned} u_t^\epsilon (x,t)=\int _{G}\frac{1}{\epsilon ^2}K_{\epsilon }(x,y)(u^\epsilon (y,t)-u^\epsilon (x,t))\,dy, \qquad x\in \Omega \end{aligned}$$with \(u^\epsilon =g(x,t)\) for \(x\notin \Omega\). For an appropriated rescaled kernel \(K_\epsilon\), we apply the Taylor series development in Carnot groups in order to prove that the solutions \(u^\epsilon\) uniformly approximate the solution of a certain local Dirichlet problem in \(\Omega\), when \(\epsilon \rightarrow 0\).