Mathematical Sciences: Partial Differential Equations and Quasiregular Mappings

点击翻译按钮获取中文摘要

Mappings from domains in Euclidean space of any dimension into the same space are said to be quasiregular if the distortion remains bounded. This means that infinitesimal dilations remain within fixed limits in all directions and at all points. The concept is similar to that of quasiconformal mapping except that quasiregular maps are not required to be univalent. Aside from their interest as transformations which preserve reasonable geometric properties, they also occur as solutions of quasilinear elliptic partial differential equations involving the gradient of scalar-valued functions. The first goal of this mathematical research is to determine the interior regularity of these solutions, which may fail since the gradient of the solution can vanish at interior points (in contrast to quasiconformal maps). A specific objective is to show that the gradients of solutions are locally of bounded mean variation. This is a natural question since it has recently been shown that the gradients are in all the Lebesgue spaces (locally). Additional work will focus on the differential operator known as the p-Laplacian which is receiving considerable attention at this time. Gradients of solutions of the corresponding homogeneous equation have their oscillation bounded by the maximum of their length. This is not the best possible estimate on the oscillation - a better one has been found in two dimensions. Efforts will be made to extend the sharper bound to higher dimensions. In a more geometric vein, work will continue on the question of boundary limits of quasiregular mappings. At issue is the extent to which a quasiregular mapping or a solution of the p-Laplacian can be expected to approach a limiting value as the independent variable approaches the boundary of the domain of definition in a nontangential manner. When the domain is a ball, the existence of nontangential limits is known to exist in sets of low Hausdorff measure in the case of the p-Laplacian, but it is not known whether quasiregular maps must have any such limits at all. Those with smooth distortion are understood, but quasiregular maps do not always have smooth or even continuous distortion.