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 $\epsilon \rightarrow 0$.