Mathematical Sciences: The Degree of Regularity of Quasiregular Mapping and Minima of Related NondifferentiableFunctions in the Calculus of Variations

数学科学：变分法中拟正则映射的正则度和相关不可微函数的极小值

• 批准号: 8807924
• 负责人: Tadeusz Iwaniec
• 金额: $ 4万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-07-01 至 1990-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8807924&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Degree; Regularity; Quasiregular

Work to be carried out on this project will focus on quasiconformal and quasiregular mapping of Euclidean space. These mappings - (the distinction being that quasiconformal maps are univalent) - arise in a number of analytical and geometric contexts. They play important roles in our understanding of classes of partial differential equations. The problems to be investigated are closely related, motivated by both theoretical interest as well as by their applicability to regularity theory for elliptic equations and estimates of singular integrals. Quasiregular mappings are known to be solutions of n- dimensional Beltrami systems of partial differential equations, which may have discontinuous terms. One of the objectives of this work will be to find minimizers for the total energy among suitable classes of these mappings. The variational problems associated with this minimization arise in nonlinear elastostatics. It has been recognized that the correct way of measuring the degree of regularity of a quasiregular mapping as well as studying the mimima of non-differentiable functionals is through the Holder exponent or by the exponent of integrability of their derivatives. A second objective of this work will be to identify the best exponents or at least describe the quantitative character of the allowable exponents. Related work will seek to find sharp estimates for two-dimensional Hilbert transforms and to examine the asymptotic behavior of the power norms of the transform.

该项目的工作将侧重于 欧氏空间的拟共形映射和拟正则映射。 这些映射-（区别在于拟共形映射 是单价的）-出现在一些分析和几何 contexts. 它们在我们理解 一类偏微分方程 存在的问题 调查是密切相关的，由理论和 兴趣以及它们对正则性理论适用性 椭圆方程和奇异积分的估计。 已知拟正则映射是n- 维Beltrami系统的偏微分方程， 其可以具有不连续的项。 的目标之一 这项工作将是找到最小的总能量， 这些映射的合适类。 变分问题 与这种最小化相关的非线性 弹性静力学 人们已经认识到，正确的测量方法 拟正则映射的正则度以及 研究不可微泛函的mimima是通过 的保持器指数或指数的可积性， 衍生物. 这项工作的第二个目标是确定 最好的指数或者至少描述了 允许指数的特征。 相关工作将力求 找到二维希尔伯特变换的精确估计， 研究幂范数的渐近行为， 变换