Mathematical Sciences: Partial Differential Equations and Quasiregular Mappings

数学科学：偏微分方程和拟正则映射

• 批准号: 8901524
• 负责人: Juan Manfredi
• 金额: $ 3.88万
• 依托单位: University of Pittsburgh
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-07-01 至 1991-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8901524&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Partial; Differential; Equations

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.

任意维欧氏空间中域的映射 进入同一个空间的都说是拟正则的，如果 仍然有界。 这意味着无限小的膨胀仍然存在 在所有方向和所有点的固定范围内。 的 概念类似于拟共形映射，除了 拟正则映射不要求是单叶的。 除了 他们的兴趣是保持合理的转换 几何性质，它们也出现作为解决方案的拟线性 椭圆型偏微分方程 标量值函数 这个数学的第一个目标 研究是为了确定这些内在规律性， 解决方案，这可能会失败，因为解决方案的梯度可以 在内点消失（与拟共形映射相反）。 一个具体的目标是表明，梯度的解决方案， 是局部有界平均变差。 这是一个自然 问题，因为最近已经表明，梯度 在所有的Lebesgue空间中（局部）。 额外的工作将集中在微分算子 被称为p-Laplacian， 注意在这个时候。 解的导数 相应的齐次方程振动有界 最大的长度。 这不是最好的可能 估计的振荡-一个更好的一个已经发现，在两个 尺寸. 将努力扩大更严格的约束， 更高的维度 从几何学的角度来看， 拟正则映射的边界极限 争论的焦点是 一个拟正则映射或 可以预期p-Laplacian接近极限值，因为 独立变量接近域的边界 以非线性的方式定义。 当域是球时， 已知非切向极限存在于集合中 在p-Laplacian的情况下，低Hausdorff测度，但它 不知道拟正则映射是否一定有这样的极限 根本 那些具有平滑失真的是可以理解的，但是 拟正则映射并不总是光滑的甚至是连续的 畸变