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。在过去的四分之一世纪里，拟正则映射被引入来描述平面和空间的变换，其变形（膨胀）对任何一点都是有界的。拟共形映射具有相同的定义，除了它们必须是一元的。拟正则映射与偏微分方程理论中的唯一性问题密切相关。本课题主要研究椭圆型和抛物型微分方程的弱解，以及整个拟正则或拟亚纯函数的分支集与其值分布之间的关系。我们还将继续研究空间中某些时变域的奇异积分和热方程。一个空间变量的情况现在被很好地理解了。目前的计划是将最近的结果推广到多维案例中。目的是研究这些域上的Dirichlet和Neumann问题，并寻求确定抛物测度相对于某射影勒贝格测度的相互绝对连续性。这将使用层势的方法，大卫累积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建立方程，不如说是建立方程。

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.