English

Harnack inequality for hypoelliptic second order partial differential operators

Analysis of PDEs 2015-09-18 v1

Abstract

We consider nonnegative solutions u:ΩRu:\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 Ω\Omega is a bounded open subset of Rn\mathbb{R}^{n} and xx denotes the point of Ω\Omega. For any fixed x0Ωx_0 \in \Omega, we prove a Harnack inequality of this type supKuCKu(x0) u \mboxs.t. Lu=0,u0,\sup_K u \le C_K u(x_0)\qquad \forall \ u \ \mbox{ s.t. } \ \mathscr{L} u=0, u\geq 0, where KK is any compact subset of the interior of the L\mathscr{L}-propagation set of x0x_0 and the constant CKC_K does not depend on uu.

Keywords

Cite

@article{arxiv.1509.05245,
  title  = {Harnack inequality for hypoelliptic second order partial differential operators},
  author = {Alessia E. Kogoj and Sergio Polidoro},
  journal= {arXiv preprint arXiv:1509.05245},
  year   = {2015}
}
R2 v1 2026-06-22T10:58:52.023Z