Clifford分析中的几类边值问题及其相关问题研究

Clifford分析边值问题研究不但能促进许多相关数学分支的发展，而且是诸多实际工程问题和物理问题的理论基础。本项目将结合已有复分析解析函数边值问题理论，利用Hardy空间分解理论、k-正则函数分解定理与轴对称域的特性等重点研究Clifford分析中旋量值函数Riemann Hilbert边值问题、k-正则函数Riemann Hilbert边值问题、矩阵边值问题及其相关问题。主要内容涉及到旋量值函数理论，Riemann Hilbert边值问题，k-正则函数性质与分解，矩阵边值问题，可再生核，Hardy空间及多Hardy空间的刻画，轴对称域上相关边值问题。在本项目中，我们将发展和开拓Clifford分析边值问题新的方法和技巧，深化数学工具，对所研究的边值问题获得若干全新的、本质性的结果，推进Clifford分析函数理论的发展。

Boundary value problems in Clifford analysis not only promote the development of many related branches of mathematics, but also are the theoretical basis of many practical engineering and physical problems. Combining the existing theory of boundary value problems for analytic function in complex analysis and using the decompositions of Hardy space, theorems of k-regular functions decomposition and characteristics of axially symmetric domain etc., we will focus on Riemann Hilbert boundary value problem for spinor-value functions, Riemann Hilbert boundary value problem for k-regular functions, matrix-valued boundary value problem and related problems in Clifford Analysis. Main content including the spinor-valued function theory, Riemann Hilbert boundary value problem, properties and decompositions of k-regular function, matrix-valued boundary value problem, reproducing kernel, characterization of Hardy space and poly-Hardy space, related boundary value problems in axially symmetric domain. In this project, we will develop and exploit new methods and techniques for boundary value problems in Clifford analysis, deepening the mathematical tools to obtain a number of new and essential results of the boundary value problem, and promoting the development of function theory in Clifford analysis.