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

9401104伊万内克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式。主要主题源于拟共形和拟正则映射中的问题。近年来，这项工作扩展到包括调和分析、变分、索博列夫空间、微分几何和拓扑学的方法。Sullivan和Donaldson关于四维流形的拟共形映射的工作给出了拟共形映射现状的主要推动力。在这一研究过程中，新的微分方程被发现，它们在许多方面推广了熟悉的柯西-黎曼系统或Beltrami方程。要研究的基本问题包括找到确保弱拟正则性蕴含强拟正则性的条件。还将研究这些映射的奇点，以及维度和可移除奇点集之间的联系。此外，还将努力分析具有某些代数结构的A-调和映射和奇异积分，例如Grassmannn或Clifford代数，目的是确定积分作为函数空间的变换时的无量纲范数。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析常常发展出近似解的方法，并估计这些近似的精度。***

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. ***