Estimates of Dirichlet Eigenvalues for a Class of Sub-elliptic Operators

Abstract

Let $\Omega$ be a bounded connected open subset in ${\mathbb{R}}^n$ with smooth boundary $\partial \Omega$. Suppose that we have a system of real smooth vector fields $X=(X_1,X_2,$ $\cdots ,X_m)$ defined on a neighborhood of $\overline{\Omega }$ that satisfies the Hörmander's condition. Suppose further that $\partial \Omega$ is non-characteristic with respect to $X$. For a self-adjoint sub-elliptic operator ${△}_X=-{\sum }_{i=1}^m{X}_i^{\ast }{X}_i$ on $\Omega$, we denote its $k^{th}$ Dirichlet eigenvalue by ${\lambda }_k$. We will provide an uniform upper bound for the sub-elliptic Dirichlet heat kernel. We will also give an explicit sharp lower bound estimate for ${\lambda }_k$, which has a polynomially growth in $k$ of the order related to the generalized Métivier index. We will establish an explicit asymptotic formula of ${\lambda }_k$ that generalizes the Métivier's results in 1976. Our asymptotic formula shows that under a certain condition, our lower bound estimate for ${\lambda }_k$ is optimal in terms of the growth of $k$. Moreover, the upper bound estimate of the Dirichlet eigenvalues for general sub-elliptic operators will also be given, which, in a certain sense, has the optimal growth order.