Mathematical Sciences: Singularities of Harmonic Function inCn

数学科学：Cn 调和函数的奇异性

• 批准号: 8819569
• 负责人: Dmitry Khavinson
• 金额: $ 3.31万
• 依托单位: University of Arkansas
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-06-01 至 1992-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8819569&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Singularities; Harmonic; Function

The main theme of this work, to investigate the holomorphic continuation of solutions of holomorphic partial differential equations in domains within the space of several real or complex varibles, will employ techniques from three areas of mathematical analysis. They are partial differential equations, complex analysis and potential theory. The extent to which solutions of elliptic equations extend from hypersurfaces to surrounding domains is believed to be directly related to the maximum extension of the Schwarz potential. It should be mentioned that the question of maximal extent of solutions of equations of hyperbolic type (wave equations) has been thoroughly investigated. In the case of equations of elliptic type, such as Laplace's equation, conjectures of what the maximal extent should be have not even been formulated. Present work will focus on very specific questions regarding homogeneous equations with holomorphic boundary values. The first objective will be to determine whether or not the maximal domain of analyticity of a solution depends only on the differential operator and the boundary, but not the specific Cauchy data. A simple device which has proved valuable in the cases of equations in two real dimensions is the Schwarz function of an analytic curve. The domain of regularity of the Schwarz function plays a crucial role in determining the maximal region of analyticity for solutions of elliptic initial value problems with analytic data. A reasonable extension of the Schwarz function has been formulated which will be examined to determine whether the basic two-dimensional results can be expanded to higher real dimensions and (any) complex ones. Preliminary analysis will be done on problems where the Cauchy data consists of polynomials, with later goals to include entire functions.

这项工作的主题是研究几个真实或复杂变量的空间内域中全态偏微分方程解决方案的全态延续，将采用数学分析的三个领域的技术。 它们是部分微分方程，复杂的分析和潜在理论。 椭圆方程的溶液从高空延伸到周围域的溶液在多大程度上与Schwarz电位的最大扩展直接相关。 应该提到的是，已经彻底研究了双曲线类型方程（波动方程）的最大解决方案解决方案的问题。 在椭圆类型方程（例如Laplace的方程式）的情况下，甚至没有对最大程度的猜想进行猜想。 目前的工作将集中于有关具有全态边界值的同质方程的非常具体的问题。 第一个目的是确定解决方案的最大分析性域是否仅取决于差分运算符和边界，而不取决于特定的cauchy数据。 一个简单的设备在两个实际维度的方程式中被证明是有价值的，这是分析曲线的Schwarz函数。 Schwarz函数规则性的领域在确定分析数据的椭圆初始值问题的解决方案的最大分析区域中起着至关重要的作用。 已经制定了Schwarz函数的合理扩展，该函数将进行检查，以确定是否可以将基本的二维结果扩展到更高的实际维度和（任何）复杂的尺寸。 初步分析将在Cauchy数据由多项式组成的问题上进行，并以后的目标包括整个功能。