Harnack inequality for hypoelliptic second order partial differential operators

Abstract

We consider nonnegative solutions $u:\Omega ⟶\mathbb{R}$ 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 $\mathcal{L}$-propagation set of $x_0$ and the constant $C_K$ does not depend on $u$.

Cited by in corpus (1)

• On Liouville-type theorems and the uniqueness of the positive Cauchy problem for a class of hypoelliptic operators