Mathematical Sciences: Singularities of Harmonic Function inCn

本工作的主题是研究全纯的 全纯偏微分解的延拓 多个真实的或复数空间内的域中的方程 变量，将采用来自数学三个领域的技术， 分析. 它们是偏微分方程，复 分析和潜力理论。 椭圆型方程解的延拓程度 从超曲面到周围域被认为是 直接关系到施瓦茨的最大扩展 潜力 应该指出的是， 双曲型方程（波）解的范围 方程）进行了深入研究。 的情况下 椭圆型方程，如拉普拉斯方程， 最大程度上应该是什么样的， 已制定。 目前的工作将侧重于非常具体的 关于全纯齐次方程的问题 边界值。 第一个目标是确定 是否是一个 解仅取决于微分算子和 边界，但不是具体的柯西数据。 一个简单的装置 这在两个真实的方程的情况下证明是有价值的 维数是解析曲线的施瓦茨函数。 的 施瓦茨函数的正则性域起着关键作用 在确定解的最大解析区域时， 椭圆型初值问题的解。 施瓦茨函数的一个合理的扩展是 将进行审查，以确定是否基本 二维结果可以扩展到更高的真实的维度 和复杂的。 初步分析将在 柯西数据由多项式组成的问题， 后期的目标包括全部功能。

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.