Mathematical Sciences: Classical Complex Analysis

申请：DMS-9532087 PI: Marshall在此资助下，Marshall将研究解析函数理论的三个领域的问题。本文将一种构造极值距离下界的良好度量的新方法应用于Ahlfors 1930的角导数问题。其次，他将研究保角映射数值计算的一种有前途的新技术的准确性。第三，他将尝试改进Dirichlet空间及其乘子的插值函数的构造，使用更自然的再现核的线性组合。保角映射作为一种工具在科学和工程中已经使用了很多年。使用保角映射的一种方法是将复杂平面上的复杂区域上的问题转换为“标准”区域上的相关问题，例如可以使用已知技术的磁盘或半平面。然后将标准区域上的解通过保角映射的逆变换为原始问题在原始区域上的解。经典地，这种方法被用来解决与拉普拉斯方程有关的问题。例如，金属薄板上的平衡温度满足拉普拉斯方程。在过去的25年里，共形映射的许多非经典应用得到了发展，其中许多都不受拉普拉斯方程的支配。例如，在电磁学、振动膜和声学、横向振动和板的屈曲、弹性、传热和流体流动等方面都有应用。虽然早期的应用使用显式解析表示来表示共形映射，但现代的应用需要更复杂的区域的共形映射，这些区域不能轻易地用初等函数表示。因此，人们必须借助于数值近似来得到期望的保角映射。马歇尔将研究一种快速计算共形映射及其逆的新技术的准确性。这种技术足够快，可以在一个典型的工作站上进行实验。他将研究的角导数问题是关于区域边界上的几何条件，保证某些共形映射（在区域内定义）在边界上的一点上扩展为共形。圆盘上的狄利克雷函数不保形，但圆盘的像具有有限的面积（计算重叠）。它们来自总能量有限的势能。马歇尔将要研究的插值问题与在薄板上构造一个温度分布密切相关，该温度分布在薄板上的某些点上具有预先设定的值，并且总能量尽可能小。

ABSTRACT Proposal: DMS-9532087 PI: Marshall Under this grant, Marshall will investigate problems in three areas of analytic function theory. A new approach to constructing good metrics for lower bounds for extremal distance will be applied to the Angular Derivative Problem of Ahlfors 1930 . Secondly, he will investigate the accuracy of a promising new technique for the numerical computation of conformal maps. Thirdly, he will attempt to improve constructions of interpolating functions for the Dirichlet space and its multipliers, using more natural linear combinations of reproducing kernels. Conformal mapping has been used as a tool in science and engineering for many years. One way conformal maps are used is to transform a problem on a complicated region in the complex plane to a related problem on a "standard" region, such as a disk or half-plane, where known techniques can be used. The solution on the standard region is then transformed by the inverse of the conformal map to a solution of the original problem on the original region. Classically, this method was used for problems related to Laplace's equation. For example, temperature at equilibrium on a thin metallic plate satisfies Laplace's equation. There have been numerous non-classical applications of conformal maps developed in the last twenty five years, many of which are not governed by Laplace's equation. There are applications in electro-magnetics, vibrating membranes and acoustics, transverse vibrations and buckling of plates, elasticity, heat transfer, and fluid flow for example. While early applications used explicit analytic representations for conformal maps, modern uses require conformal maps of more complicated regions which cannot be represented easily in terms of elementary functions. One must therefore resort to numerical approximations to the desired conformal maps. Marshall will investigate the accuracy of a new technique which rapidly computes conformal maps and their inverses. This technique is fast enough that it can be used for experimentation on a typical workstation. The angular derivative problem he will investigate is about geometrical conditions on the boundary of a region that guarantee that certain conformal maps (defined inside the region) extend to be conformal at a point on the boundary. Dirichlet functions on the disk are not conformal, but the image of the disk has finite area (counting overlaps). They arise from potentials with bounded total energy. The interpolation problems Marshall will study are closely related to constructing a temperature distribution on a thin metal plate with preassigned values at certain points on the plate, and with smallest possible total energy.