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 \(\e\ge 0\).