Mathematical Sciences: Quasiconformal and Quasiregular Mappings and Elliptic Partial Differential Equations

本基金资助博士后研究拟共形映射的边值问题和与拟正则映射相关的加权Sobolev不等式。项目中解决的问题有助于理解在分析这些映射时解析和几何方面之间的相互作用。通过拟共形映射，可以理解同一维欧氏空间之间的一元映射，其中映射的变形或扩张是有界的。准正则映射满足相同的条件，但不要求是一元的。两者都反映了复变量解析函数研究的自然结果。我们将研究拟共形映射的导数。人们一直认为这些函数总是具有局部可积的幂。事实并非如此。然而，最近一个先进的平均导数的概念表明，正是这个函数最终将有正确的幂积分（对于保角映射，这个平均和传统的导数是一致的）。也要检查的是平均导数的径向边界值在距离边界。其他工作集中在椭圆方程与拟共形和拟正则映射组成时的变化以及具有可积扩张的Sobolev映射。在后者中，最基本的问题是证明如果任意映射的导数的幂是由雅可比矩阵乘以一个有界函数有界，那么这个映射是开的和离散的。结果在二维情况下得到了验证。

Postdoctoral funding through this award provides support for mathematical research on problems concerning boundary values of quasiconformal mappings and weighted Sobolev inequalities associated with quasiregular mappings. The problems addressed in the project contribute to the understanding of the interplay between analytic and geometric aspects in the analysis of these mappings. By quasiconformal mappings one understands univalent mappings between Euclidean spaces of the same dimension in which the distortion or dilatation of the mapping is bounded. Quasiregular mappings satisfy the same conditions but are not required to be univalent. Both reflect a natural outgrowth of the study of analytic functions of a complex variable. Work will be done examining the derivative of quasiconformal mappings. It had been thought that these functions always had powers which were locally integrable. This is not the case. However, a recently advanced notion of average derivative suggests that it is this function which will ultimately have the right power integrals (for conformal maps this average and conventional derivatives agree). Also to be examined are the radial boundary values of average derivatives in terms of the distance to the boundary. Other work focuses on changes in elliptic equations when composed with quasiconformal and quasiregular maps and on Sobolev mappings with integrable dilatation. In the latter, the fundamental question is one of showing that if powers of a derivative of an arbitrary map are bounded by the Jacobian times a bounded function then the mapping is open and discrete. The results has been validated in the two-dimensional case.