The First Eigenvalue of the Kohn-Laplace Operator in the Heisenberg Group

In this paper, by extending the notions of harmonic transplantation and harmonic radius in the Heisenberg group, we give an upper bound for the first eigenvalue for the following Dirichlet problem: $$(P_{\Omega}) \left\{ \begin{array}{lllll} -\Delta_{\mathbb{H}^1} u & = & \lambda u & \mbox{in} & \Omega u & = & 0 & \mbox{on} & \partial \Omega, \end{array} \right.$$ where $ \Omega $ is a regular bounded domain of $ \mathbb{H}^1$ with smooth boundary and $\Delta_{\mathbb{H}^1}$ is the Kohn-Laplace operator. Using the results of P.Pansu which give the relation between the volume of $\Omega$ and the perimeter of its boundary. we prove the following $$ \lambda_{1}( \Omega ) \leq C_{\Omega} \displaystyle \frac{ l_{11}^2 }{ \displaystyle \max_{ \xi \in \Omega } r_{\Omega}^2(\xi)} $$ where $l_{11} $ is the first strictly positive zero of the Bessel function of first kind and order 1, $ C_{\Omega} $ is a constant depending of $ \Omega $ and $r_{\Omega}(\xi) $ is the harmonic radius of $ \Omega $ at a point $\xi$ of $\Omega.$