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 $$u_t^{\epsilon }(x,t)={\int }_G\frac{1}{{\epsilon }^2}{K}_{\epsilon }(x,y)(u^{\epsilon }(y,t)-u^{\epsilon }(x,t))dy,x\in \Omega$$ with $u^{\epsilon }=g(x,t)$ for $x\notin \Omega$. For appropriate rescaled kernel $K_{\epsilon }$ we prove that solutions $u^{\epsilon }$, when $\epsilon \to 0$, 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.

20 pages