報告題目：A few sharp estimates of harmonic functions with applications to Steklov eigenfunctions
報告人：張城博士，清華大學
報告時間：2025年1月7上午10:00-11:00
報告地點：6号教學樓101
報告摘要：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.
報告人簡介：張城博士，清華大學數學中心助理教授。研究方向為調和分析，目前主要研究興趣是流形上的特征值和特征函數，以及函數的零點等。2014年本科畢業于浙江大學數學系，2019年博士畢業于美國約翰霍普金斯大學。主持國家海外優青項目，國家自然科學基金面上項目，國家重點研發計劃青年科學家項目。在J. Math. Pures, Appl. Adv. Math., Anal. PDE, J. Funct. Anal., Comm. Math. Phys., Camb. J. Math.等高水平學術期刊發表論文十餘篇。
