Mathematical Sciences: Singularities of Harmonic Function inCn

这项工作的主要主题是研究全纯偏微分方程在几个实数或复数变量的域内解的全纯延拓，将采用三个数学分析领域的技术。它们是偏微分方程、复变分析和势理论。椭圆方程的解从超曲面扩展到周围区域的程度被认为与Schwarz势的最大扩展直接相关。值得一提的是，双曲型方程（波动方程）解的最大范围问题已经得到了深入的研究。在椭圆型方程的情况下，例如拉普拉斯方程，关于最大范围应该是什么的猜想甚至还没有被表述出来。目前的工作将集中在具有全纯边值的齐次方程的非常具体的问题上。第一个目标是确定解的可解析性的最大域是否仅取决于微分算子和边界，而不是特定的柯西数据。解析曲线的施瓦兹函数是一个简单的方法，它在二维实空间的方程中被证明是有价值的。对于具有解析数据的椭圆型初值问题的解，Schwarz函数的正则性域在确定其可解析性的极大区域中起着至关重要的作用。对Schwarz函数进行了合理的扩展，并将对其进行检验，以确定基本二维结果是否可以扩展到更高的实维和（任何）复维。初步分析将在柯西数据由多项式组成的问题上完成，之后的目标是包括整个函数。

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.