Maximum Principle in the Unbounded Domain of Heisenberg Type Group

Abstract

In this paper, through polar coordinates and constructing an auxiliary function we prove the maximum principle in the unbounded domain $\Omega$ as following: suppose that there exists $u(\xi ) \in C(\overline{\Omega } )$ bounded above, solution of

$$ \left\{ {\begin{array}{*{20}c} {Lu(\xi ) + c(\xi )u(\xi ) \ge 0 \, \xi \in\Omega ,{\text{ c(}}\xi )\le 0,} \\ {u(\xi ) \le 0 \, \xi \in \partial\Omega ,} \\ \end{array} } \right. $$

then $u(\xi ) \le 0$ in $\Omega$. Here $\Omega$ is an open connected subset of Heisenberg type group $G$ such that the following condition holds: there exists $\xi_{0} \in G$ such that $\overline{{\xi_{0} \circ\Omega }}$ lies on one side of an hyperplane parallel to $y_{1}$ axis.