Harnack inequality for hypoelliptic second order partial differential operators

TL;DR

This paper establishes a Harnack inequality for nonnegative solutions of second order hypoelliptic PDEs, providing bounds within the propagation set that are independent of the specific solution.

Contribution

It proves a Harnack inequality for hypoelliptic operators, extending classical results to a broader class of second order PDEs with variable coefficients.

Findings

Harnack inequality holds for solutions of hypoelliptic equations

Constant in inequality is independent of the solution

Applicable within the operator's propagation set

Abstract

We consider nonnegative solutions $u:\Omega\longrightarrow \mathbb{R}$ of second order hypoelliptic equations \begin{equation*} \mathscr{L} u(x) =\sum_{i,j=1}^n \partial_{x_i} \left(a_{ij}(x)\partial_{x_j} u(x) \right) + \sum_{i=1}^n b_i(x) \partial_{x_i} u(x) =0, \end{equation*} where $\Omega$ is a bounded open subset of $\mathbb{R}^{n}$ and $x$ denotes the point of $\Omega$. For any fixed $x_0 \in \Omega$, we prove a Harnack inequality of this type $$\sup_K u \le C_K u(x_0)\qquad \forall \ u \ \mbox{ s.t. } \ \mathscr{L} u=0, u\geq 0,$$ where $K$ is any compact subset of the interior of the $\mathscr{L}$-propagation set of $x_0$ and the constant $C_K$ does not depend on $u$.