北航数学论坛学术报告

--- 分析、偏微分方程与动力系统讨论班(2025春季第13讲)

A few sharp estimates of harmonic functions with applications to Steklov eigenfunctions

张城（清华大学）

时间：2025年06月19日（周四）下午16:00-17:00

地点：学院路老主楼105

摘要: On smooth compact manifolds with smooth boundary, we first establish the sharp lower bounds for the restrictions of harmonic functions in terms of their frequency functions, by  using a combination of microlocal analysis and frequency function techniques by Almgren and Garofalo-Lin. The lower bounds can be saturated by Steklov eigenfunctions on Euclidean balls and a family of symmetric warped product manifolds. Moreover, as in Sogge and Taylor, we analyze the interior behavior of harmonic functions by constructing a parametrix for the Poisson integral operator and calculate its composition with the spectral cluster. By using microlocal analysis, we obtain several sharp estimates for the harmonic functions whose traces are quasimodes on the boundary. As applications, we establish the almost-orthogonality, bilinear estimates and transversal restriction estimates for Steklov eigenfunctions, and discuss the numerical approximation of harmonic functions. This is joint work with Xing Wang (Hunan U.)

报告人简介: 张城，现任清华大学数学中心助理教授，国家级人才。2014年本科毕业于浙江大学，2019年博士毕业于美国约翰霍普金斯大学,2019-2022任罗切斯特大学客座助理教授。研究方向是调和分析及其应用，特别是流形上的特征值与特征函数问题，获得国家级项目资助，论文发表在Camb.J. Math., CMP, Adv.Math., JMPA, APDE, TAMS等著名期刊。

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