报告题目：A few sharp estimates of harmonic functions with applications to Steklov eigenfunctions
报告人：张城博士，清华大学
报告时间：2025年1月7上午10:00-11:00
报告地点：6号教学楼101
报告摘要：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. Asapplications, we establish the almost-orthogonality, bilinear estimates and transversal restriction estimates for Steklov eigenfunctions, and discuss the numerical approximation of harmonic functions.
报告人简介：张城博士，清华大学数学中心助理教授。研究方向为调和分析，目前主要研究兴趣是流形上的特征值和特征函数，以及函数的零点等。2014年本科毕业于浙江大学数学系，2019年博士毕业于美国约翰霍普金斯大学。主持国家海外优青项目，国家自然科学基金面上项目，国家重点研发计划青年科学家项目。在J. Math. Pures, Appl. Adv. Math., Anal. PDE, J. Funct. Anal., Comm. Math. Phys., Camb. J. Math.等高水平学术期刊发表论文十余篇。
