We consider nonnegative solutions $u:Omegalongrightarrow mathbb{R}$ of second order hypoelliptic equations egin{equation*} mathscr{L} u(x) =sum_{i,j=1}^n partial_{x_i} left(a_{ij}(x)partial_{x_j} u(x) ight) + 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 $$sup_K u le C_K u(x_0)qquad orall u mbox{ s.t. } mathscr{L} u=0, ugeq 0,$$ where $K$ is any compact subset of the interior of the $mathscr{L}$-{it propagation set of $x_0$} and the constant $C_K$ does not depend on $u$. The research that led to the present paper was partially supported by a grant of the group GNAMPA of INdAM
Harnack Inequality for Hypoelliptic Second Order Partial Differential Operators
KOGOJ, ALESSIA ELISABETTA;
2016
Abstract
We consider nonnegative solutions $u:Omegalongrightarrow mathbb{R}$ of second order hypoelliptic equations egin{equation*} mathscr{L} u(x) =sum_{i,j=1}^n partial_{x_i} left(a_{ij}(x)partial_{x_j} u(x) ight) + 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 $$sup_K u le C_K u(x_0)qquad orall u mbox{ s.t. } mathscr{L} u=0, ugeq 0,$$ where $K$ is any compact subset of the interior of the $mathscr{L}$-{it propagation set of $x_0$} and the constant $C_K$ does not depend on $u$. The research that led to the present paper was partially supported by a grant of the group GNAMPA of INdAMI documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
