Mathematical Sciences: Regularity Problems in Nonlinear Potential Theory and Quasiregular Mappings

该项目继续进行数学研究，旨在利用拟正则映射的经典理论与变分方程的非线性椭圆系统之间新发现的关系。 拟正则映射最初是作为复函数理论的一个适度分支（拟共形映射）出现的。 这些映射的几何特征在于它们将无穷小球体带入无穷小椭球体。 在本世纪下半叶，人们发现这些映射被证明是几何学中的基本对象，例如泰希米勒理论以及其他分析领域。 目前的工作重点是唐纳森-沙利文在拟共形四流形上的工作如何产生适用于非线性偏微分方程问题的新技术。 这项工作的首要目标之一是了解在偶数维空间和奇数维空间上定义的拟正则映射的研究中出现的区别（如果有的话）。 一些更深层次的新发现只有在偶维情况下才能知道。 基于 Hodge 分解和 Cacciappoli 型不等式的论证可能会产生有关奇数维的附加信息。 拟正则映射的第二个应用涉及由元素为 Riesz 变换的矩阵组成的奇异积分变换的向量值类似物。 人们相信这些算子的 p 范数不依赖于底层空间的维度。尽管可能不可能获得规范的精确值，但我们将努力证明界限是无量纲的。 为了实现这一目标，引入了一种称为复杂旋转方法的新技术，该技术适用于广泛的积分算子来估计其映射范数。 最后，更多几何性质的工作将继续解决拟正则映射的可移除集的最大维数以及豪斯多夫维数在此类映射下如何扭曲的问题。

This project continues mathematical research aimed at exploiting newly discovered relationships between the classical theory of quasiregular mappings and nonlinear elliptic systems of variational equations. Quasiregular mappings first appeared as a modest branch of complex function theory (quasiconformal mapping). The mappings are characterized geometrically by the property that they carry infinitesimal spheres into infinitesimal ellipsoids. In the second half of this century it was discovered that these mappings proved to be fundamental objects in geometry, such a Teichmuller theory, as well as to other areas of analysis. The present work focuses on how the Donaldson - Sullivan work on quasiconformal four-manifolds has led to new techniques applicable to questions in nonlinear partial differential equations. One of the first objectives of this work is to understand what, if any, distinctions arise in the study of quasiregular mappings defined on even and odd dimensional spaces. Some of the deeper new discoveries are only known in the even dimensional case. Arguments based on Hodge decompositions and Cacciappoli type inequalities may yield additional information about odd dimensions. A second application of quasiregular mappings relates to vector-valued analogues of singular integral transformations consisting of matrices whose elements are Riesz transforms. It is believed that the p-norms of these operators do not depend on the dimension of the underlying space. Although it may be impossible to achieve exact values for the norms, work will be done in showing that the bounds are dimension-free. To achieve this, a new technique called the complex method of rotation has been introduced which applies to a broad class of integral operators in estimating their mapping norms. Finally work of a more geometric nature will continue on the questions of the maximum dimension of removable sets for quasiregular mapping and how Hausdorff dimension is distorted under such maps.