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

这个项目继续进行数学研究，旨在利用新发现的准正则映射经典理论和非线性椭圆型变分方程组之间的关系。拟正则映射最初是作为复变函数论(拟共形映射)的一个温和分支出现的。这种映射的几何特征是它们将无穷小球面带入无穷小椭球体。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，例如泰希穆勒理论，以及其他分析领域。本文主要研究了Donaldson-Sullivan在拟共形四维流形上的工作如何产生了适用于非线性偏微分方程组问题的新技术。这项工作的首要目标之一是理解在奇数维空间和偶数维空间上定义的拟正则映射的研究中出现的区别。一些更深层次的新发现只有在偶次元情况下才知道。基于Hodge分解和Cacciappoli型不等式的论点可能会产生关于奇数维的额外信息。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模拟。人们认为这些算子的p-范数不依赖于基础空间的维度。虽然可能不可能获得规范的精确值，但我们将努力证明边界是无量纲的。为了实现这一点，引入了一种新的技术，称为复数旋转方法，它适用于一类广泛的积分算子来估计其映射范数。最后，关于拟正则映射的可去集的最大维度以及Hausdorff维度在这种映射下如何扭曲的问题，将继续进行更具几何性质的工作。

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.