Quasiconformal Mappings, Harmonic Analysis and Nonlinear Elasticity from the Prospective of PDEs

偏微分方程视角下的拟共形映射、调和分析和非线性弹性

• 批准号: 9706611
• 负责人: Tadeusz Iwaniec
• 金额: $ 14.57万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706611&HistoricalAwards=false
• 关键词: Quasiconformal; Mappings; Harmonic; Analysis; Nonlinear

Iwaniec ABSTRACT Iwaniec will pursue questions about quasiregular mappings which arise as generalizations of geometric aspects of analytic functions of one complex variable. The mappings in question solve important first order systems of PDEs analogous in many respects to the Cauchy-Riemann equations. The solutions of these systems can be viewed as "absolute" minimizers of certain energy functionals. It is striking how tight the connection is between quasiregular mappings and the recent development of the nonlinear elasticity theory whose mathematical principles were already formulated by S.S. Antman and J. Ball in 1976-77. Roughly speaking, the theory of elasticity studies mappings, referred to as deformations of elastic bodies, which minimize the so-called stored energy functionals. These functionals are not always convex and the deformations need not be quasiconformal but the governing PDEs are much the same. The Jacobian determinant, in particular, has been subjected to a great deal of investigation. Ths project accounts for this study together with some generalizations concerning wedge products of closed differential forms. Quasiconformal mappings generalize to higher dimensions the concept of analytic functions that is so fruitful for science and engineering in two space dimensions. Trying to extend this theory to three or more space dimensions proved quite difficult and the mathematics of such extension is being developed now. Iwaniec will work on how to extend familiar concepts in the plane, like degree, to similar concepts in space. The theory of non linear elasticity, among others, will benefit from this developments.

伊万尼克 摘要 Iwaniec将探讨有关拟正则映射的问题， 作为解析函数的几何方面的推广而出现 一个复杂的变量。映射问题解决重要的一阶系统的偏微分方程类似的柯西-黎曼方程在许多方面。这些系统的解可以看作是某些能量泛函的“绝对”极小化。令人惊讶的是，拟正则映射与非线性弹性理论的最新发展之间的联系是多么紧密，非线性弹性理论的数学原理已经由S.S.安特曼 和J.鲍尔在1976-77年。粗略地说，弹性理论 研究映射，称为弹性体的变形， 其最小化所谓的储能泛函。这些 泛函并不总是凸的，并且变形不需要是 但支配偏微分方程是相同的。的 特别是雅可比行列式， 调查的交易。 本项目是本研究的一部分 以及一些关于楔积的推广 闭微分形式 拟共形映射将解析函数的概念推广到高维空间，这在二维空间的科学和工程中是如此富有成果。试图将这一理论扩展到三个或更多的空间维度证明是相当困难的，这种扩展的数学现在正在发展。Iwaniec将致力于如何将平面中熟悉的概念（如度）扩展到空间中的类似概念。非线性弹性理论将从这些发展中受益。