Mathematical Sciences: Quasiconformal and Quasiregular Mappings and Elliptic Partial Differential Equations

数学科学：拟共形和拟正则映射以及椭圆偏微分方程

• 批准号: 9305742
• 负责人: Frederick Gehring
• 金额: $ 3.68万
• 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: 1993
• 资助国家: 美国
• 起止时间: 1993-05-01 至 1996-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9305742&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Quasiconformal; Quasiregular; Mappings

Postdoctoral funding through this award provides support for mathematical research on problems concerning boundary values of quasiconformal mappings and weighted Sobolev inequalities associated with quasiregular mappings. The problems addressed in the project contribute to the understanding of the interplay between analytic and geometric aspects in the analysis of these mappings. By quasiconformal mappings one understands univalent mappings between Euclidean spaces of the same dimension in which the distortion or dilatation of the mapping is bounded. Quasiregular mappings satisfy the same conditions but are not required to be univalent. Both reflect a natural outgrowth of the study of analytic functions of a complex variable. Work will be done examining the derivative of quasiconformal mappings. It had been thought that these functions always had powers which were locally integrable. This is not the case. However, a recently advanced notion of average derivative suggests that it is this function which will ultimately have the right power integrals (for conformal maps this average and conventional derivatives agree). Also to be examined are the radial boundary values of average derivatives in terms of the distance to the boundary. Other work focuses on changes in elliptic equations when composed with quasiconformal and quasiregular maps and on Sobolev mappings with integrable dilatation. In the latter, the fundamental question is one of showing that if powers of a derivative of an arbitrary map are bounded by the Jacobian times a bounded function then the mapping is open and discrete. The results has been validated in the two-dimensional case.

通过该奖项提供博士后资助， 边值问题的数学研究 拟共形映射与加权Sobolev不等式 与拟正则映射相关。 所探讨问题的 该项目有助于理解 在分析这些问题时， 映射。 通过拟共形映射，我们可以理解单叶映射 同维欧氏空间之间的映射，其中 映射的变形或扩张是有界的。 拟正则映射满足相同的条件，但不是 必须是单价的。 两者都反映了 对复变函数的研究。 工作 我们将研究拟共形映射的导数。 人们一直认为，这些职能总是有权力， 是局部可积的 事实并非如此。 但 最近提出的平均导数概念表明， 这个函数最终会有正确的力量 积分（对于共形映射，此平均和常规 衍生品）。 也要检查的是径向边界 平均导数的值，以到 边界 其他工作集中在椭圆方程的变化 当由拟共形映射和拟正则映射组成时， 具有可积扩张的Sobolev映射。 在后者中， 一个基本的问题是，如果一个人的权力 任意映射的导数都有界于雅可比时间 一个有界函数，那么这个映射是开的和离散的。 的 结果已在二维情况下得到验证。