A Subelliptic Analogue of Aronson-Serrin's Harnack Inequality

2011/09/21 by Luca Capogna, Capogna, Luca, Giovanna Citti +3
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1109.4596

arxiv created 2012/12/29 · arxiv updated 2013/01/01

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=(X₁,...,X_m) in in \Om× (0,T) with \Om ⊂ M an open subset of a manifold M with control metric d corresponding to X and a measure dσ follows from the basic hypothesis of doubling condition and a weak Poincaré inequality. We also show that such hypothesis hold for a class of Riemannian metrics g_\e collapsing to a sub-Riemannian metric lim_{\e→ 0} g_\e=g₀ uniformly in the parameter \e≥ 0.

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