Mathematical Sciences: Quasiconformal Mappings and NonlinearPotential Theory

拟共形映射是从平面映射的研究中发展而来的，这些平面映射的无穷小畸变保持在固定的范围内。这种变换被认为是保角映射的自然数学推广（它将圆无限小地映射到圆上）。该理论在几个方向和维度上得到了发展，与非线性势理论和黎曼曲面的Teichmuller理论有重要的联系。本课题研究退化椭圆型偏微分方程的拟共形映射和解的几何性质问题。这项工作源于最近的一项发现，即映射到球上的拟共形同胚具有稳定性：在定域的某些子集中，它们的畸变是全局控制的，而不考虑定域的几何形状。关于可以映射到球上的域，人们知之甚少；将做一些工作来描述这些域。这种现象最初是在共形图中观察到的，随后产生了几个深层结果。第二项研究涉及p-拉普拉斯方程的超解的性质。这些函数构成了非线性势理论的基础。两个特定的目标是研究可能的精细拓扑结构，这些拓扑结构可以产生此类函数的最佳连续性结果，并寻求除球以外区域的边界哈纳克原理。

Quasiconformal mappings evolved from studies of plane mappings whose infinitesimal distortions remained bounded within fixed limits. Such transformations were considered as the natural mathematical generalization of a conformal map (which infinitesimally maps circles onto circles). The theory has grown in several directions and dimensions, making important contact with nonlinear potential theory and Teichmuller theory of Riemann surfaces. This project is concerned with problems relating quasiconformal maps and the geometric behavior of solutions to degenerate elliptic partial differential equations. The work derives from a recent discovery that quasiconformal homeomorphisms mapping onto a ball have a stability property: in certain subsets of the domain, their distortion is globally controlled without regard to the geometry of the domain. Little is understood concerning domains which may be mapped onto a ball; work will be done to characterize such domains. This phenomenon was first observed in conformal maps where several deep results has subsequently been produced. A second line of investigation concerns the properties of supersolutions of the p-Laplace equation. These functions form the basis for a nonlinear potential theory. Two particular goals the study of possible fine topologies available which yield the best continuity results for such functions and to seek a boundary Harnack principle for domains other than a ball.