A Subelliptic Analogue of Aronson-Serrin's Harnack Inequality

TL;DR

This paper extends the Harnack inequality to degenerate parabolic PDEs associated with Lipschitz vector fields on manifolds, demonstrating it holds under doubling and Poincaré conditions, including for collapsing Riemannian metrics.

Contribution

It establishes a subelliptic analogue of Aronson-Serrin's Harnack inequality for a broad class of degenerate PDEs on manifolds with collapsing metrics.

Findings

Harnack inequality derived under doubling and Poincaré conditions

Applicable to PDEs with Lipschitz vector fields on manifolds

Holds for collapsing Riemannian metrics converging to sub-Riemannian limits

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 $X=(X_1,...,X_m)$ in in $\Om\times (0,T)$ with $\Om \subset M$ an open subset of a manifold $M$ with control metric $d$ corresponding to $X$ and a measure $d\sigma$ follows from the basic hypothesis of doubling condition and a weak Poincar\'e inequality. 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 $\e\ge 0$.