A subelliptic analogue of Aronson–Serrin’s Harnack inequality

Abstract

We study the Harnack inequality for weak solutions of a class of degenerate parabolic quasilinear PDE $$\begin{aligned} \partial _t u={-}X_i^* A_i(x,t,u,Xu)+ B(x,t,u,Xu), \end{aligned}$$ in cylinders $$\Omega \times (0,T)$$ where $$\Omega \subset M$$ is an open subset of a manifold $$M$$ endowed with control metric $$d$$ corresponding to a system of Lipschitz continuous vector fields $$X=(X_1,\ldots ,X_m)$$ and a measure $$d\sigma $$ . We show that the Harnack inequality follows from the basic hypothesis of doubling condition and a weak Poincare inequality in the metric measure space $$(M,d,d\sigma )$$ . We also show that such hypothesis hold for a class of Riemannian metrics $$g_\epsilon $$ collapsing to a sub-Riemannian metric $$\lim _{\epsilon \rightarrow 0} g_\epsilon =g_0$$ uniformly in the parameter $$\epsilon \ge 0$$ .