Deformations of Finite n-Harmonic Energy

有限n次谐波能量的变形

• 批准号: 0701059
• 负责人: Jani Onninen
• 金额: $ 11万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701059&HistoricalAwards=false
• 关键词: Deformations; Finite; Harmonic; Energy

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.

这个项目的特点是在欧氏n-空间域之间的映射的新类。它们产生于最近的研究变分积分，出现在非线性弹性和作为一个综合的想法，从多维几何函数理论，特别是从拟共形几何。最近，人们对将复调和函数的经典理论扩展到高维空间产生了相当大的兴趣。分析和几何函数理论的连接支持了越来越多的信心，在欧几里德n-空间域的n-调和变形代表了一个更好的推广全纯函数比经典调和映射。分析中出现n-谐波变形有几种自然的方式。其中之一是由Antman、Ball和Ciarlet开创的非线性弹性数学模型。为了用简单的术语来描述它，弹性理论研究物质在给定区域上的变形，变形的构型使所谓的储能泛函最小化。经典Teichmuller理论，其极值映射之间的黎曼曲面，提供了一个类似的观点。广义地说，这个理论关注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尽量减少积分范数的内部失真，这反过来又减少到研究的n-调和映射。第一个自然的问题，无论是从理论和实践的角度来看，是是否n-调和能量是有限的，如果是这样，它是否假设一个最小值之间的所有同胚。这两个问题都是相当困难的，因为没有边界条件施加在映射上，到目前为止只有零碎的答案。人们可能期望能量最小化器是非内射的。然而，主要的研究者已经发现，在自然的附加假设下，极小元是同胚的。这里的关键是Iwaniec和他在研究n次谐波极小化器时开发的技术。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，这种趋势变得更加明显，并导致理论和应用数学家越来越努力地放弃极端的概括和抽象概念。这个项目，它位于几何函数理论和非线性弹性的接口，是与这些努力。它肯定会与数学和物理学的许多其他领域建立联系，其中一些已经存在。这将需要认真的合作研究。最后，它将扩大大学生参与研究活动的范围。