Deformations of Finite n-Harmonic Energy

这个项目的特点是在欧几里得n空间的域之间的映射的新类别。它们源于最近对非线性弹性中出现的变分积分的研究，并作为多维几何函数理论，特别是准共形几何思想的综合。近年来，人们对将复调和函数的经典理论扩展到更高的维度产生了相当大的兴趣。与几何函数理论的分析和联系，支撑了一个日益增长的信心，即欧几里得n空间域的n调和变形比经典调和映射更好地代表了全纯函数的推广。在分析中有几种自然的方式产生n谐波变形。其中之一是由Antman、Ball和Ciarlet开创的非线性弹性数学模型。用简单的术语来说，弹性理论研究的是材料体在给定域上的变形，即使所谓的储能泛函最小化的变形构型。经典的Teichmuller理论，其黎曼曲面之间的极值映射，提供了一个类似的视角。广义地说，这个理论关注的是具有最小可能上模的失真函数。主要研究者的方法是最小化内部畸变的积分范数，这反过来又减少了对n谐波映射的研究。从理论和实践的角度来看，第一个自然的问题是n-调和能量是否是有限的，如果是，它是否在所有同胚中假设一个最小值。这两个问题都相当困难，因为没有对映射施加边界条件，到目前为止只有不完整的答案。有人可能会认为能量最小化是非内射的。然而，首席研究员发现，在自然附加假设下，最小值是同胚的。这里的关键是Iwaniec和他在研究n谐波最小化时开发的技术。在这些研究中，分析和几何拓扑之间的相互作用是至关重要的。一般的现代数学，特别是数学分析，其发展越来越依赖于物理和几何直觉。近年来，这种趋势变得更加明显，并导致纯粹数学家和应用数学家更加努力地放弃极端的概括和抽象概念。这个项目，位于几何函数理论和非线性弹性的界面，与这些努力是一致的。它肯定会与数学和物理学的许多其他领域建立联系，其中一些已经到位。这将需要认真的合作研究。最后，它将扩大本科生对研究活动的参与。

This project features new classes of mappings between domains in Euclidean n-space. They grew out of recent studies of the variational integrals that arise in nonlinear elasticity and as a synthesis of ideas from multidimensional geometric function theory, in particular, from quasiconformal geometry. There has been considerable interest recently in extending the classical theory of complex harmonic functions to higher dimensions. Analysis and connections with geometric function theory underpin a growing confidence that n-harmonic deformations of domains in Euclidean n-space represent a better generalization of holomorphic functions than classical harmonic mappings. There are several natural ways that n-harmonic deformations arise in analysis. One of those is the mathematical model of nonlinear elasticity pioneered by Antman, Ball, and Ciarlet. To describe it in simplistic terms, the theory of elasticity studies deformations of a material body onto a given domain, the deformed configuration that minimizes the so-called stored energy functional. Classical Teichmuller theory, with its extremal mappings between Riemann surfaces, offers a similar perspective. Broadly speaking, this theory is concerned with a distortion function with the smallest possible supremum norm. The principal investigator's approach is to minimize the integral-norms of the inner distortion, which in turn reduces to the study of n-harmonic mappings. The first natural questions, both from the theoretical and practical points of view, are whether the n-harmonic energy is finite and, if so, whether it assumes a minimum value among all homeomorphisms. Both of these questions are quite difficult, because no boundary conditions are imposed on the mappings, and only fragmentary answers exist thus far. One might expect that the energy minimizers would be noninjective. The principal investigator has discovered, however, that under natural additional assumptions the minimizers are homeomorphisms. The key here is the technique that Iwaniec and he developed when studying n-harmonic minimizers. In these studies the interplay between analysis and geometric topology is crucial.Modern mathematics in general and mathematical analysis in particular rely more and more for their development on physical and geometric intuition. In recent years this trend has become more pronounced and has led to increased efforts by pure and applied mathematicians to abandon extreme generalizations and abstract concepts. This project, which lies at the interface of geometric function theory and nonlinear elasticity, is aligned with those efforts. It will certainly establish connections with many other areas of mathematics and with physics, some of which are already in place. It will entail serious collaborative research. Finally, it will broaden the participation of undergraduates in research activities.