Mathematical Sciences: Quasiconformal Mappings and NonlinearPotential Theory

数学科学：拟共形映射和非线性势理论

基本信息

批准号：
8902749
负责人：
Frederick Gehring
金额：
$ 3.42万
依托单位：
Regents of the University of Michigan - Ann Arbor
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：
1989-06-15 至 1992-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=8902749&HistoricalAwards=false
关键词：
Mathematical Sciences Quasiconformal Mappings NonlinearPotential

项目摘要

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.

准文献映射是从对无限扭曲的平面映射的研究演变而来的，其无穷小变形在固定极限内保持界限。这种转换被认为是保形图的自然数学概括（无限量映射到圆圈上）。该理论已经在多个方向和维度上发展，与非线性潜在理论和黎曼表面的Teichmuller理论建立了重要联系。该项目涉及有关准文化图和解决方案的几何行为而与椭圆形偏微分方程退化的几何行为有关的问题。这项工作来自最近的发现，即映射到球上的准形式同构具有稳定性：在域的某些子集中，它们的失真在全局控制不考虑域的几何形状。关于可以映射到球上的域的了解很少。将完成此类域的特征。这种现象首先是在随后产生的几个深层结果中观察到的。第二次调查涉及p-laplace方程的超溶液的特性。这些功能构成了非线性潜在理论的基础。两个特定的目标研究可用的可能的精细拓扑结构，这些拓扑可为此类功能带来最佳的连续性结果，并为除球以外的其他域寻求边界Harnack原理。