Constructing discrete harmonic functions in wedges

The essential spectrum of the Neumann--Poincare operator on a domain with corners

Exploiting the homogeneous structure of a wedge in the complex plane, we compute the spectrum of the anti-linear Ahlfors-Beurling transform acting on the associated Bergman space. Consequently, the similarity equivalence between the Ahlfors-Beurling transform and the Neumann-Poincare operator provides the spectrum of the latter integral operator on a wedge. A localization technique and conformal mapping lead to the first complete description of the essential spectrum of the Neumann-Poincare operator on a planar domain with corners, with respect to the energy norm of the associated harmonic fie…

调和函数的几个精确估计及其在 Steklov 特征函数中的应用

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

本文结合微局部分析与频率函数技术，建立了光滑紧流形上调和函数限制的精确下界，并通过构造参数化分析其内部行为，最终应用于 Steklov 特征函数的估计与调和函数的数值逼近。

A note on constructing sharp examples for $L^{p}$ norms of eigenfunctions and quasimodes near submanifolds

In this note we analyse $L^{p}$ estimates for Laplacian eigenfunctions and quasimodes and their associated sharp examples. In particular we use previously determined estimates to produce a new set of estimates for restriction to thickened neighbourhoods of submanifolds. In addition we produce a family flat model quasimode examples that can be used to determine sharpness of estimates on Laplacian eigenfunctions restricted to subsets. For each quasimode in the family we show that there is a corresponding spherical harmonic that displays the same growth properties. Therefore it is enough to check…