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

Abstract

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:

$$\begin{aligned} (P_{\Omega }) \left\{ \begin{array}{rllll} -\Delta _{\mathbb {H}^1} u &{} = &{} \lambda u &{}\quad \text{ in } &{} \Omega \\ u &{} = &{} 0 &{}\quad \text{ on } &{} \partial \Omega , \end{array} \right. \end{aligned}$$

where $\Omega$ is a regular bounded domain of $\mathbb {H}^1$ and $\Delta _{\mathbb {H}^1}$ is the Kohn–Laplace operator. Using a result of Pansu which gives a relation between the volume of $\Omega$ and the perimeter of its boundary, we prove that

$$\begin{aligned} \lambda _{1}( \Omega ) \le C_{\Omega } \frac{ l_{11}^2 }{ \max _{ \xi \in \Omega } r_{\Omega }^2(\xi )} \end{aligned}$$

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 $\Omega .$