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

[401104] Iwaniec这个项目所代表的数学研究的一般领域是非线性偏微分方程。主要的主题来自于拟共形映射和拟正则映射的问题。近年来，这项工作扩大到包括谐波分析、变分法、索博列夫空间、微分几何和拓扑学等方法。Sullivan和Donaldson关于四流形的拟共形映射的研究，对拟共形映射的发展起到了重要的推动作用。在这一研究过程中，人们发现了新的微分方程，它们在许多方面推广了我们所熟悉的柯西-黎曼系统或贝尔特拉米方程。要研究的基本问题包括寻找保证弱拟正则性蕴涵强拟正则性的条件。我们还将研究这些映射的奇异性，以及维度和可移动奇异集之间的联系。额外的努力将用于分析A-调和映射和奇异积分，它们带有某些代数结构，如grassmann或Clifford代数，其目标是确定作为函数空间变换的积分上的无维范数。偏微分方程是物理世界数学建模的基础。数学分析的作用与其说是创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。＊＊＊

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. ***