图 书 馆
拉普拉斯本征函数的零集Zero sets of Laplace eigenfunctions |
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| 课程网址: | https://videolectures.net/8ecm2021_logunov_zero_sets/ |
| 主讲教师: | Alexandr Logunov |
| 开课单位: | 8ECM会议 |
| 开课时间: | 2021-07-06 |
| 课程语种: | 英语 |
| 中文简介: |  |
| 课程简介: | In the beginning of 19th century Napoleon set a prize for the best mathematical explanation of Chladni’s resonance experiments. Nodal geometry studies the zeroes of solutions to elliptic differential equations such as the visible curves that appear in Chladni’s nodal portraits. We will discuss the geometrical and analytic properties of zero sets of harmonic functions and eigenfunctions of the Laplace operator. For harmonic functions on the plane there is an interesting relation between local length of the zero set and the growth of harmonic functions. The larger the zero set is, the faster the growth of harmonic function should be and vice versa. Zero sets of Laplace eigenfunctions on surfaces are unions of smooth curves with equiangular intersections. Topology of the zero set could be quite complicated, but Yau conjectured that the total length of the zero set is comparable to the square root of the eigenvalue for all eigenfunctions. We will start with open questions about spherical harmonics and explain some methods to study nodal sets, which are zero sets of solutions of elliptic PDE. |
| 关 键 词: | 拉普拉斯本征函数; 节点几何; 椭圆微分方程 |
| 课程来源: | 视频讲座网 |
| 数据采集: | 2024-05-23：liyq |
| 最后编审: | 2024-05-23：liyq |
| 阅读次数: | 1 |