报告题目： Estimates of Dirichlet Eigenvalues for a Class of Sub-elliptic Operators

报告人: 陈化教授 (武汉大学)

时间：2019年12月20日上午 10:30-11:30

地点：浙江大学玉泉校区工商楼200-9

摘要:

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 H\"{o}rmander'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 M\'{e}tivier index. We will establish an explicit asymptotic formula of $\lambda_{k}$ that generalizes the M\'{e}tivier'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.

报告人简介：陈化, 武汉大学教授，博士生导师。现为武汉大学数学协同创新中心主任，国务院学科数学评议组成员，湖北省暨武汉数学会理事长，湖北省计算科学省重点实验室主任。陈化的研究方向为偏微分方程的微局部分析理论，退化型偏微分方程，具生物和医学背景的偏微分方程和偏微分方程的谱理论；至今已主持国家自然科学基金项目18项，其中包括国家杰出青年基金，参加八五、九五、十一五国家重点项目，并主持十二五、十三五国家重点项目以及国家基金委天元基金交叉平台项目等，还为国家重大项目973核心数学项目组成员并获国家教育部跨世纪优秀人才基金。曾获国家教育部科技进步二等奖两次，2017年主持的科研项目获得国家教育部自然科学奖一等奖。

All are welcome!

联系人：张挺 (zhangting79@zju.edu.cn)

浙江大学数学科学学院