关于几类调和拟共形映射性质的研究

Due to the close association among harmonic mappings, quasiconformal mappings, partial differential equations etc, quasiconformal harmonic mappings have become a research hotspot. This project plans to consider several classes of partial differential equations with quasiconformal mappings, and study it Lipschitz continuity etc. Concrete research contents are as follows: (1) We investigate the Lipschitz continuity, the function space attribute problems of the solutions and quasiconformal solutions of partial differential equations related to harmonic mappings by means of harmonic mapping theory, quasiconformal mapping theory, function space theory and the techniques of partial differential equation and functional analysis. (2) By using subordinate theory and subharmonic function theory, we will consider the Bohr type inequality problems related to two conjectures proposed by Ponnusamy. The research of this project will improve the related research of harmonic mappings, and make harmonic mappings more closely related to other fields. Therefore, it is very significant in theory.

调和映射与拟共形映射、偏微分方程等研究领域的紧密相联,使得调和拟共形映射成为当前的研究热点.本项目计划结合拟共形映射,考虑几类与调和映射相关的偏微分方程,研究其解的Lipschitz连续等性质.具体研究内容如下:(1)综合调和映射理论、拟共形映射理论、函数空间理论以及偏微分方程和泛函分析中的技巧,研究与调和映射相关的偏微分方程的解以及拟共形映射解的Lipschitz连续性、函数空间属性问题等.(2)利用从属理论和次调和函数理论,研究Bohr型不等式问题,着重考虑Ponnusamy提出的两个相关的猜测.本项目的研究将更加完善调和映射的相关研究,也会使得调和映射与其它相关领域的结合研究更加紧密,因此具有重要的理论意义.

此项目研究期间,我们按原计划对与调和映射相关的偏微分方程的解或者拟共形映射解的一些性质展开了研究,得到了系列结果,共发表SCI论文(标注本项目资助)5篇,含接收SCI论文1篇.具体如下：. 本项目主要研究了α-调和方程、非齐次双调和方程以及非齐次Yukawa方程等方程的解的一些性质,例如Lipschitz连续性、Schwarz型引理、Landau型定理以及与Bergman型空间的关系等问题.主要结果发表在J. Geom. Anal.、Complex Var. Elliptic、Hokkaido Math. J.、Anal. Math. Phys.国际权威刊物上.