题目： On Eigenvalues Problems for Sub-elliptic Operators

报告人：陈化教授

（武汉大学）

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 Hormander's condition.Suppose further that $/partial/Omega$ is non-characteristic with  respect to $X$.  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 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 Metivier index. We will establish an explicitasymptotic formula of $/lambda_{k}$ that generalizes the Metivier'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.

时间：5月16日（周四）下午14:00-15：00

地点：565net必赢客户端本部教二楼 527 教室

联系人：牛冬娟

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