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

Iwaniec抽象Iwaniec将探讨关于拟正则映射的问题，这些映射是由一个复变量解析函数的几何方面的推广而产生的。所讨论的映射求解重要的一阶偏微分方程组，在许多方面类似于柯西-黎曼方程。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如此紧密，其数学原理已经由S.S.Antman和J.Ball在1976-77年间提出。粗略地说，弹性理论研究的是映射，即弹性物体的变形，它使所谓的储能泛函最小化。这些泛函并不总是凸的，变形也不一定是准共形的，但支配偏微分方程组几乎是相同的。尤其是雅可比行列式，已经受到了大量的研究。这个项目说明了这一研究以及关于闭微分形式的楔积的一些推广。拟共形映射将解析函数的概念推广到更高的维度，在二维空间中，解析函数的概念在科学和工程中都是卓有成效的。试图将这一理论扩展到三维或更多空间维度被证明是相当困难的，这种扩展的数学现在正在发展中。Iwaniec将致力于如何将平面上熟悉的概念，如度，扩展到空间中的类似概念。除其他外，非线性弹性理论将受益于这一发展。

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.