关于调和映照的边界特征及其相关问题的研究

Harmonic mappings, as the generalization of conformal mappings, have more flexible properties and better applicability in high-dimensional space. It plays a crucial role in many areas of mathematics, physics, and engineering. In recent years, on the studies of geometric properties, boundary functions and some estimates of harmonic mappings have been active topics in the field of complex analysis. In this project, we mainly study the following problems: (1). By using Poisson formula of harmonic mappings, we will study the relationship between the boundary function and some integrable function classes, and then obtain the relationship between its derivative and H^p space. By estimating the Jacobian and Schwarz type derivative, we will study the quasiconformality and Lipschitz property of high-dimensional harmonic functions and obtain some sharp constants estimating. (2). For different measure, we will study the Schwarz lemma and Heinz inequality of harmonic mappings and their applications. (3). We will study extremal problems of harmonic mappings and the Landau type theorem, Bloch constant for non-univalent harmonic mappings. Our results will improve and enrich the theorems of harmonic mappings and quasiconformal mappings, and have important research significance. The results of this project will be appeared by publishing papers. 5-7 papers are expected to be published in some important international journals of China or in abroad.

作为共形映照的推广，调和映照具有更加灵活的性质，在高维空间中具有更好的适用性并在数学、物理及工程学的诸多领域中发挥着重要的作用。近年来，研究调和映照的几何特性、边界特征、偏差估计等已成为复分析领域较为活跃的研究热点之一。本项目中我们主要研究以下问题：（1）利用调和映照的Poisson表示研究其边界与某些可积函数类的关系，进而得到其导数与H^p空间的关系。通过估计Jacobian与Schwarz型导数来研究高维调和映照的拟共形性质和Lipschitz性质及其精确的常数估计。（2）在不同的度量下研究调和映照的Schwarz引理、Heinz不等式估计及其应用。（3）研究调和映照的极值问题，探索非单叶调和映照的Landau型定理、Bloch常数等问题。我们的结果将拓广和丰富调和映照理论和拟共形映照理论，具有重要的研究意义。项目的研究成果将以论文的形式出现，预期在国内外重要刊物上发表5-7篇论文。

调和映照与拟共形映照是几何函数论中较为活跃的研究方向，并对复动力系统、Teichmuller空间、Klein群等产生重要的影响。本项目中我们主要研究了如下内容：（1）、调和映照的边界特征及其拟共形延拓。我们得到了解析部分为凸的单位圆盘到一般区域（有界或无界域）上的调和映照成为拟共形映照的充分必要条件。这一结果推广了Pavlovic和Kalaj的经典结果。（2）、Poisson方程解的边界Schwarz引理。我们得到了比调和映照更一般的函数族：Poisson方程的解的渐进精确的边界Schwarz引理。特别地，若该族函数具有拟共形性质，我们的结果是与经典的结果是渐进一致的。（3）、我们得到了单位球上调和函数的Harnack不等式估计。这一结果是精确的且推广了前人的结果。目前为止，已经发表了13篇与该项目相关的文章。