Harnack estimates for degenerate parabolic equations modeled on the subelliptic p-Laplacian

TL;DR

This paper proves a Harnack inequality for a class of degenerate quasi-linear PDEs modeled on subelliptic p-Laplacian operators, extending previous results to more general geometric and measure-theoretic settings.

Contribution

It generalizes Harnack estimates to settings involving Lipschitz vector fields, non-smooth measures, and various geometric structures, broadening applicability of prior results.

Findings

Harnack inequality established for subelliptic p-Laplacian models.

Results apply to Riemannian manifolds with non-negative Ricci curvature.

Extends to measures with Muckenhoupt weights.

Abstract

We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype {equation*} \partial_tu= -\sum_{i=1}^{m}X_i^\ast (|\X u|^{p-2} X_i u){equation*} where $p\ge 2$, $ \ \X = (X_1,..., X_m)$ is a system of Lipschitz vector fields defined on a smooth manifold $\M$ endowed with a Borel measure $\mu$, and $X_i^*$ denotes the adjoint of $X_i$ with respect to $\mu$. Our estimates are derived assuming that (i) the control distance $d$ generated by $\X$ induces the same topology on $\M$; (ii) a doubling condition for the $\mu$-measure of $d-$metric balls and (iii) the validity of a Poincar\'e inequality involving $\X$ and $\mu$. Our results extend the recent work in \cite{DiBenedettoGianazzaVespri1}, \cite{K}, to a more general setting including the model cases of (1) metrics generated by H\"ormander vector fields and Lebesgue measure; (2) Riemannian manifolds with non-negative Ricci curvature and Riemannian volume forms; and (3) metrics generated by non-smooth Baouendi-Grushin type vector fields and Lebesgue measure. In all cases the Harnack inequality continues to hold when the Lebesgue measure is substituted by any smooth volume form or by measures with densities corresponding to Muckenhoupt type weights.