复与超复分析中边值问题与奇异积分方程

项目名称： 复与超复分析中边值问题与奇异积分方程

项目编号： No.10871150

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 金属学与金属工艺

项目作者： 杜金元

作者单位： 武汉大学

项目金额： 27万元

中文摘要： 本课题《复与超复分析中的边值问题和奇异积分方程》包括两个平台上的工作。(1)单复分析中解析与多解析函数以及多调和函数的边值问题和奇异积分方程，(2)超复分析中正则与高阶正则函数以及高阶调和函数的边值问题和奇异积分方程。在这两工作方向上，研究者们均做出了创新性的工作，不少工作具有原创性。两个方向上的工作各具特点并有侧重，又相互影响和促进,形成了一个有机整体。 研究者们一方面运用单复变中经典解析函数边值问题和奇异积分方程基本理论和成果为导向，寻求取值在Clifford代数上的正则函数的广义解析函数的相应问题，大面积形成了高维上的成果。另一方面，后高维上的结果使得人们更能洞见经典问题中本质所在，又回过来拓广和精化经典单复变中的结果，形成推广深化，产生不少交叉性工作点和创新性成果。例如, 刺激多解析函数以及多调和函数的边值问题研究得到极大发展, 进而又影响到高阶正则函数和高阶调和函数的边值问题的研究。本课题的成果能够应用和影响到其它许多学科分支，如弹性力学、复偏微分方程、多复分析以及物理学方面的一些分支。因此,本课题研究成果既具很好理论价值又具广阔应用前景。

中文关键词： 解析函数; 正则函数; Clifford分析; 边值问题; 奇异积分方程

英文摘要： The project titled "The Boundary Value Problems and Singular Equations in Complex and Hypercomplex Analysis" includes the works in two frameworks. [1] Boundary value problems and singular integral equations for analytic and polyanalytic functions as well as harmonic functions in classical complex variable; [2] boundary value problems and singular integral equations for regular and higher order regular functions as well as hypercomplex harmonic functions of higher order in Clifford analysis. In these two areas the researchers have new works with creativity and some are original. The research in these two areas can mutually influence and form a unified whole. For example, the research results in the former area, i.e., the boundary value problems and the singular integral equations in the classical complex analysis, shed light to the corresponding problems of the hypercomplex functions in the Clifford algebra and a lot results in higher dimensional case are gotten. On the other hand, the result of the later work, i.e., the hypercomplex analysis, can essentially offer insights into the methods of deepening and refining the corresponding results in the classical complex analysis, forming new research directions and growing point. The research results can be applied to many fields of science, such as partial differential equation, complex variables and some areas of physics. Consequently this research topic has deep theoretic importance and wide application prospective as well.

英文关键词： Analysis functions; Regular function; Clifford analysis; Boundary value problem; Singular integral equation