Mathematical Sciences: Partial Differental Equations and Systems Related to Quasiregular Mappings

本项目将考虑与欧几里得空间中至少3维的拟正则映射相关的偏微分方程理论中的问题，试图将经典函数理论的某些方面推广到高维。欧几里得空间中的潜在势理论是非线性的，调和函数被准正则映射下一定意义上不变的拟线性方程的解所取代。拟共形映射非常适合用解析方法通过这些拟线性椭圆方程来研究。它们在合成下是封闭的，这使得它们在拓扑学和几何学中非常有用。这些研究对于非牛顿流体和金属蠕变的数学建模具有重要意义。将寻求进一步的联系，使应用科学和纯数学受益。

This project will consider problems in the theory of partial differential equations related to quasiregular mappings in Euclidean space of dimension at least 3, in an attempt to extend to higher dimensions some aspects of classical function theory. The underlying potential theory in Euclidean space is nonlinear and harmonic functions are replaced by solutions of certain quasilinear equations which are invariant in a certain sense under quasiregular mappings. Quasiconformal mappings are well suited to be studied with analytic methods via these quasilinear elliptic equations. They are closed under composition, making them very useful in Topology and Geometry. These studies are important in the mathematical modelling of non-newtonian fluids and creep of metals. Further connections to benefit applied science as well as pure mathematics will be sought.