调和函数水平集的几何性质

调和函数是一类非常重要而又相当特殊的函数，通过研究调和函数往往能够为理解一般的椭圆偏微分方程提供新的看法与技术。本项目主要关心调和函数水平集的局部几何性质，即调和函数的指定的某个水平集的几何性质。希望解决的关键科学问题是：（1）单位圆盘上调和函数过原点的水平线在原点处曲率的最佳上界估计；（2）维数>=3时单位球上调和函数过原点的水平集在原点处曲率的上界估计；（3）单位圆盘上调和函数过原点的水平线长度的绝对上界估计。

Harmonic functions are among the most important and special functions. The study of them can usually provide us new views and techniques to understand the general elliptic partial differential equations. In this project, we mainly concern the local geometric properties of the level sets of harmonic functions, i.e. the geometric properties of a given level set of a harmonic function. We hope to solve the following three problems: (1) sharp upper bound estimates of the curvature at the origin of the level curve through the origin of a harmonic function on the unit disk; (2) upper bound estimates of the curvature at the origin of the level set through the origin of a harmonic function on the unit ball when the dimension >=3; (3) absolute upper bound estimates of the length of level curve through the origin of a harmonic function on the unit disk.