Mathematical Sciences: Regularity Problems for Variational Integrals and Quasiregular Mappings

在这个项目中有两个主要主题需要遵循。其中一个涉及涉及拟正则映射的正则性问题。这些是平面或更高维空间的映射，其中以可测量的方式控制小集合(如盘)的失真。这些映射不要求是单叶的(拟共形)，它们被刻画为(退化类型的)椭圆型偏微分方程组的解。拟正则映象研究的主要问题之一是寻找这类映象增长的估计，以及刻画满足给定边界条件的极值映象。在两个维度中，一个最终会导致复杂的希尔伯特变换。各种正则性结果依赖于这种变换的范数的估计，特别是p次方范数。将努力提高对这一标准的估计。第二行的研究将集中在驻点为准共形的函数类的变分能量积分。具体地说，将根据全域上的能量积分乘以子集的度量的幂来估计给定域的子集上的能量积分。权力的最佳价值仍有待确定。目前，唯一已知的结果是在区域是实心圆盘或球的情况下。

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.