# PDE Seminar

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

### 陈化 教授 (武汉大学)

#### 2019年02月22日13:30-14:30 闵行校区数学楼102报告厅

Abstract: Let $\Omega$ be a bounded connected open subset in $\mathbb{R}^n$ with a smooth boundary $\partial\Omega$. Given the systems of real smooth vector fields $X=(X_{1},X_{2}, \cdots, X_{m})$ defined on a neighborhood of $\overline{\Omega}$, which satisfying the H\"{o}rmander's condition, and $\partial\Omega$ is non-characteristic for $X$. For a self-adjoint sub-elliptic operator $\triangle_{X}= -\sum_{i=1}^{m}X_{i}^{*} X_i$ on $\Omega$, we denote its $k^{th}$ Dirichlet eigenvalue by $\lambda_k$. We obtain an uniform upper bound for the sub-elliptic Dirichlet heat kernel, and then we give an explicit sharp lower bound estimate of $\lambda_{k}$ which is polynomial increasing in $k$ with the order relating to the generalized M\'{e}tivier index. Furthermore, we establish an explicit asymptotic formula of $\lambda_{k}$ which generalize the M\'{e}tivier's results in 1976. This asymptotic formula implies that under a certain condition our lower bound estimate for $\lambda_{k}$ is optimal in sense of the order of $k$. On the other hand, the upper bound estimates of Dirichlet eigenvalues for general sub-elliptic operators are also given, which in some sense will be precise from the result of this talk.