Mathematical Sciences: Regularity Problems for Variational Integrals and Quasiregular Mappings

数学科学：变分积分和拟正则映射的正则问题

• 批准号: 9007946
• 负责人: Tadeusz Iwaniec
• 金额: $ 3.5万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-08-01 至 1993-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9007946&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Regularity; Problems; Variational

There are two main themes to be followed in this project. One concerns questions of regularity involving quasi-regular mappings. These are mappings of the plane or higher dimensional space in which distortion of small sets such as discs is controlled in a measurable way. The maps are not required to be univalent (quasi-conformal) and they are characterized as solutions of systems of elliptic partial differential equations (of degenerate type). One of the primary concerns in the study of quasi-regular mappings is that of finding estimates on the growth of such mappings as well as characterizing extremal mappings subject to given boundary conditions. In two dimensions, one is eventually led to the complex Hilbert transform. Various regularity results depend on estimates of the norm of this transform, specifically the p-th power norm. Work will be done on sharpening estimates for the norm. A second line of investigation will focus on variational energy integrals of classes of functions whose stationary points are quasi-conformal. In particular, work will be done in estimating energy integrals over subsets of a given domain in terms of the energy integral over the full domain, multiplied by a power of the measure of the subset. The best value of the power remains to be determined. At this time, the only known results are in the case where the domain is a solid disc or ball.

这个项目有两个主要的主题。一个是涉及准正则映射的正则性问题。这些是平面或高维空间的映射，其中小集合（如圆盘）的畸变以可测量的方式控制。映射不需要是一元的（拟共形的），它们被表征为椭圆型偏微分方程（退化型）系统的解。在拟正则映射的研究中，一个主要的问题是找到对这种映射的增长的估计，以及在给定的边界条件下描述极值映射的特征。在二维空间中，最终会得到复希尔伯特变换。各种规律性结果依赖于这个变换的范数的估计，特别是p次幂范数。将开展工作，提高对标准的估计。研究的第二条线将集中于一类函数的变分能量积分，这些函数的平稳点是拟共形的。特别地，我们要做的工作是估计给定域的子集上的能量积分用整个域上的能量积分来表示，乘以子集测度的幂。这种权力的最佳价值还有待确定。此时，唯一已知的结果是在域是实心圆盘或球的情况下。