学术报告(陈化 1.21)

发布人:周妍 发布日期:2019-01-17
活动时间
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报 告 人: 陈化 教授 (武汉大学)

地  点:新数学楼416报告厅

摘  要:  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.

 

数学学院

2019年1月17日