A Subelliptic Analogue of Aronson-Serrin's Harnack Inequality

Abstract

We show that the Harnack inequality for a class of degenerate parabolic quasilinear PDE \p_t u=-X_i^* A_i(x,t,u,Xu)+ B(x,t,u,Xu), associated to a system of Lipschitz continuous vector fields $X=(X_1,...,X_m)$ in in \Om\times (0,T) with \Om \subset M an open subset of a manifold $M$ with control metric $d$ corresponding to $X$ and a measure $d\sigma$ follows from the basic hypothesis of doubling condition and a weak Poincar\'e inequality. We also show that such hypothesis hold for a class of Riemannian metrics g_\e collapsing to a sub-Riemannian metric \lim_{\e\to 0} g_\e=g_0 uniformly in the parameter \e\ge 0