Deformations of Finite n-Harmonic Energy

这个项目的特点是欧几里得n-空间中域之间的新映射类。它们源于最近对非线性弹性力学中出现的变分积分的研究，并作为来自多维几何函数理论，特别是来自准共形几何的思想的综合。最近，人们对将经典的复调和函数理论扩展到更高的维度有很大的兴趣。分析和与几何函数理论的联系支持了一个日益增长的信心，即欧氏n-空间中区域的n-调和变形比经典调和映射代表了更好的全纯函数的推广。在分析中，有几种自然的方式产生n次调和变形。其中之一是由Antman、Ball和Ciarlet首创的非线性弹性数学模型。用简单的术语来描述它，弹性理论研究的是物质物体在给定区域上的变形，即使所谓的存储能量泛函最小化的变形构型。经典的TeichMuller理论及其在黎曼曲面之间的极值映射提供了类似的观点。广义地讲，这个理论涉及的是具有最小可能上确界范数的失真函数。主要研究者的方法是最小化内失真的积分范数，这反过来又归结为对n-调和映射的研究。从理论和实践的角度来看，第一个自然的问题是n-调和能量是否有限，如果是，它是否在所有同胚中假定一个最小值。这两个问题都很困难，因为映射上没有施加任何边界条件，而且到目前为止只存在零散的答案。人们可能会认为，能量减小器将是非注射的。然而，首席研究者发现，在自然的附加假设下，极小化子是同胚的。这里的关键是Iwaniec和他在研究n-调和极小化时开发的技术。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。现代数学，特别是数学分析，越来越依赖于物理和几何直觉的发展。近年来，这一趋势变得更加明显，并导致纯粹的和应用的数学家更加努力地放弃极端的概括和抽象的概念。该项目位于几何函数理论和非线性弹性力学的交界处，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，其中一些领域已经到位。这将需要认真的合作研究。最后，它将扩大本科生参与研究活动的范围。

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.