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) {-Delta(H1) u = lambda u in Omega u = 0 on phi Omega, where Omega is a regular bounded domain of H-1 and Delta(H1) is the Kohn-Laplace operator. Using a result of Pansu which gives a relation between the volume of O and the perimeter of its boundary, we prove that lambda(1)(Omega) <= -C-Omega -l(11)(2)/ma(chi epsilon Omega) r(Omega)(2)(xi) where l(11) is the first strictly positive zero of the Bessel function of first kind and order 1, CO is a constant depending of Omega, and r(Omega)(xi) is the harmonic radius of Omega at a point xi of Omega.