Analysis&PDE |Estimates of Dirichlet Eigenvalues for a Class of Sub-elliptic Operators
报告人: 陈化教授 (武汉大学)
地点:浙江大学玉泉校区工商楼200-9
时间:2019.12.20 8:30-12:00
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。
联系人:张挺 (zhangting79@zju.edu.cn)
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