Mathematical Sciences: Quasiconformal Analysis and Harmonic Integrals with Applications to Nonlinear Elasticity

数学科学：拟共形分析和调和积分及其在非线性弹性中的应用

• 批准号: 9401104
• 负责人: Tadeusz Iwaniec
• 金额: $ 13.5万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-05-01 至 1997-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401104&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Quasiconformal; Analysis; Harmonic

9401104 Iwaniec The general area of mathematical research represented by this project is that of nonlinear partial differential equations. The main themes grew out of problems in quasiconformal and quasiregular mappings. The work expanded in recent years to include methods from harmonic analysis, calculus of variations, Sobolev spaces, differential geometry and topology. A major impetus to the current state of quasiconformal mapping was given by work of Sullivan and Donaldson on quasiconformal mappings of four-manifolds. In the course of this research, new differential equations were discovered which in many ways generalize the familiar Cauchy-Riemann system or the Beltrami equation. Basic questions to be studied include that of finding conditions thatensure weakly quasiregularity implies strong quasiregularity. Work will also be done investigating singularities of these mappings and the connection between dimension and removable sets of singularities. Additional efforts will be made to analyze A- harmonic mappings and singular integrals which carry certain algebraic structures, such as Grassmannn or Clifford algebras with a goal of determining dimension-free norms on the integrals when treated as transformations of function spaces. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates onthe accuracy of these approximations. ***

9401104 iWaniec该项目代表的数学研究的一般领域是非线性偏微分方程。 主要主题是出于准文献和准磁映射的问题而出现的。 近年来，这项工作扩展到包括谐波分析，变化的计算，Sobolev空间，差异几何形状和拓扑的方法。 沙利文（Sullivan）和唐纳森（Donaldson）在四元式映射的四个manifolds的准文字映射方面给出了对当前准文化映射状态的主要动力。 在这项研究的过程中，发现了新的微分方程，从许多方面概括了熟悉的Cauchy-Riemann系统或Beltrami方程。 要研究的基本问题包括发现弱的准牙条性的条件意味着很强的准杂志。 还将完成这些映射的奇异性以及维度和可移动奇点集之间的联系。 将采取额外的努力来分析A-谐波映射和奇异积分，这些积分具有某些代数结构，例如Grassmannn或Clifford代数，其目的是确定作为功能空间变换时积分上的无维规范。 部分微分方程构成了物理世界数学建模的基础。 数学分析的作用并不是创建方程，而是提供有关解决方案的定性和定量信息。 这可能包括有关独特性，光滑和成长的问题的答案。 此外，分析通常会开发用于近似解决方案和对这些近似值准确性的估计的方法。 ***