A subelliptic analogue of Aronson-Serrin’s Harnack inequality
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open in viewerWe 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 Poincaré 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 $\e\ge 0$.
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- 4/19/13
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- 2020-09-22
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