## Mathematical Sciences Faculty Publications

Article

4-19-2013

#### Publication Title

Mathematische Annalen

#### Abstract

We study the Harnack inequality for weak solutions of a class of degenerate parabolic quasilinear PDE \p_t u=-X_i^* A_i(x,t,u,Xu)+ B(x,t,u,Xu), in cylinders $\Om\times (0,T)$ with $\Om \subset M$ 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.,,,.X_m)\ and a measure dσ. We show that the Harnack inequality follows from the basic hypothesis of doubling condition and a weak Poincar\'e inequality in the metric measure space (M, d, dσ). 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 $\e\ge 0$.

357

3

1175

1198

#### DOI

10.1007/s00208-013-0937-y

Preprint