Mathematical Sciences: Partial Differential Equations and Quasiregular Mappings

从欧几里得空间的任意维域到同一空间的映射，如果畸变保持有界，则称为拟正则。这意味着无穷小的膨胀在所有方向和所有点上都保持在固定的范围内。这个概念类似于拟共形映射，除了拟正则映射不需要是一元的。除了作为保持合理几何性质的变换之外，它们还可以作为涉及标量函数梯度的拟线性椭圆偏微分方程的解出现。这项数学研究的第一个目标是确定这些解的内部规律性，这可能会失败，因为解的梯度可以在内部点消失（与拟共形映射相反）。一个具体的目标是证明解的梯度是局部有界平均变化的。这是一个很自然的问题，因为最近已经证明梯度存在于所有勒贝格空间（局部）。额外的工作将集中在被称为p-拉普拉斯算子的微分算子上，它目前受到相当大的关注。相应齐次方程的梯度解的振荡以其长度的最大值为界。这并不是对振动最好的估计——在二维空间中已经发现了一个更好的估计。将努力把更尖锐的界限扩展到更高的维度。在更几何的脉络中，将继续研究拟正则映射的边界限制问题。问题在于，当自变量以非切的方式接近定义域的边界时，拟正则映射或p-拉普拉斯算子的解在多大程度上可以期望接近一个极限值。当定义域是球时，已知在p-拉普拉斯的低Hausdorff测度集合中存在非切极限，但不知道拟正则映射是否必须有这样的极限。具有平滑畸变的映射是可以理解的，但是准正则映射并不总是具有平滑甚至连续畸变。

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.