Special Session 38: 

Harnack Inequality in sub-Riemannian settings

Alessia Kogoj
University of Urbino "Carlo Bo"
Italy
Co-Author(s):    Sergio Polidoro
Abstract:
We consider nonnegative solutions $u:\Omega\longrightarrow \mathbb{R}$ of second order hypoelliptic equations $$ \mathcal{L} u(x) =\sum_{i,j=1}^n \partial_{x_i} \left(a_{ij}(x)\partial_{x_j} u(x) \right) + \sum_{i=1}^n b_i(x) \partial_{x_i} u(x) =0,$$ 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 $$\sup_K u \le C_K u(x_0)\qquad \forall \ u \ \mbox{ such that } \ \mathcal{L} u=0, u\geq 0,$$ 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 $u$. \ The result presented are obtained in collaboration with Sergio Polidoro.