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

通过该奖项提供的博士后资助为关于拟共形映射的边值问题和与拟正则映射相关的加权索博列夫不等式的数学研究提供了支持。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，人们理解了相同维欧氏空间之间的单叶映射，其中映射的扭曲或伸缩是有界的。拟正则映射满足相同的条件，但不要求是单叶的。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的幂总是局部可积的。事实并非如此。然而，最近提出的平均导数概念表明，这个函数最终将具有正确的幂积分(对于保角映射，这个平均导数和传统导数是一致的)。还将检查平均导数的径向边界值，该值与边界的距离有关。其他工作集中在椭圆型方程在由拟共形和拟正则映象组成时的变化，以及具有可积伸缩的Sobolev映象。在后者中，基本问题是证明如果任意映射的导数的幂由雅可比乘以有界函数有界，则该映射是开的和离散的。计算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.