Document Type

Article

Publication Date

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 $\e\ge 0$.

Volume

357

Issue

3

First Page Number

1175

Last Page Number

1198

DOI

10.1007/s00208-013-0937-y

Version

Preprint

Rights

Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.

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