Mathematical Sciences: Quasiregular Mappings and the Heat Equation

本项目继续对拟正则映射和热方程进行数学研究。 在过去的世纪中，拟正则映射被引入到描述关于任意点的变形（伸缩）是有界的平面和空间的变换。拟共形映射具有相同的定义，只是要求它们是单叶的。拟正则映射与偏微分方程理论中的唯一性问题密切相关。本计画将集中于椭圆型与抛物型微分方程式之弱解之研究，以及整拟正则或拟亚纯函数之分支集与其值分布之关系。工作也将继续对奇异积分和热方程的某些时间变化域的空间。单空间变量的情况现在已经很好地理解了。目前的计划是将最近的结果推广到多层面的情况。目的是研究这些区域上的Dirichlet和Neumann问题，并试图确定抛物测度关于某个射影Lebesgue测度的相互绝对连续性。这将使用层势法、大卫建立方案和奇异积分估计。偏微分方程是物理世界数学建模的基础。数学分析的作用与其说是创造方程，不如说是创造方程。

This project continues mathematical research on quasiregular mappings and the heat equation. Quasiregular mappings were introduced during the past quarter century to describe transformations of the plane and space whose distortion (dilatation) about any point is bounded. Quasiconformal maps have the same definition except that they are required to be univalent. Quasiregular maps are closely related to questions of uniqueness in the theory of partial differential equations. This project will focus on investigations into weak solutions of elliptic and parabolic differential equations and the relationship between the branch set of an entire quasiregular or quasimeromorphic function and its value distribution. Work will also continue on singular integrals and the heat equation for certain time varying domains in space. The one space variable case is now well understood. Current plans are to carry forward recent results into the multidimensional case. The goals are to study the Dirichlet and Neumann problems on these domains and seek to determine the mutual absolute continuity of parabolic measure with respect to a certain projective Lebesgue measure. This will use the method of layer potentials, the David buildup scheme and singular integral estimates. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is.