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

该项目继续进行数学研究，旨在 利用新发现的古典 拟正则映射理论与非线性椭圆型方程组 变分方程 拟正则映射最初是作为 复变函数论的一个小分支（拟共形 映射）。 映射的几何特征是： 性质，他们进行无穷小领域到无穷小 椭圆体 在本世纪的后半叶， 这些映射被证明是几何学中的基本对象， 这样的Teichmuller理论，以及其他领域的分析。 本工作的重点是如何唐纳森-沙利文 在拟共形四维流形上的工作导致了新的技术 适用于非线性偏微分问题 方程 这项工作的首要目标之一是 了解什么，如果有的话，区别出现在研究 定义在奇、偶维空间上的拟正则映射。 一些更深层次的新发现只有在晚上才知道 维度案例 基于Hodge分解和 Cacciappoli型不等式可以产生额外的信息 奇怪的维度 拟正则映射的第二个应用涉及： 奇异积分变换的向量值模拟 由其元素是Riesz变换的矩阵组成。 它 我们认为这些算子的p范数不依赖于 底层空间的维度。尽管它可能是 不可能达到规范的精确值，工作将 证明了边界是无量纲的。 实现 这是一种新的技术，称为复杂的旋转方法， 它适用于广泛的一类积分 算子估计其映射范数。 最后，更多的几何性质的工作将继续在 可去集的最大维数问题 拟正则映射与Hausdorff维数的畸变 在这样的地图下。

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.