Extremal Problems in Quasiconformal Geometry and Nonlinear PDEs, an Invitation to n- Harmonic Hyperelasticity

在这个项目中，主要研究者开始研究有限（通常是无界）失真映射的极值问题，希望为几何函数理论和非线性椭圆偏微分方程的进一步发展提供背景。攻击计划依赖于几何和物理直觉，特别是从非线性弹性理论和材料科学中得出的。这代表了近年来变得更加明显的一种趋势，并且它已经导致理论数学家和应用数学家越来越多地努力将这些想法和结果联合收割机结合起来。该项目的中心是有限变形映射和超弹性映射之间的显着关系（首先由首席研究员和Jani Onninen设想）。这两种理论都受变分原理和令人信服的共同数学兴趣的问题。在几何函数论中已经建立的新的前沿包括畸变函数的多凸积分和相关的总n-谐波能量。特别感兴趣的是最小平均失真的映射，其存在性，正则性和全局可逆性与深未解决的数学问题有关。例如，目前的建议采取了困难的问题，有关变形的球面环在欧几里德n-空间，和一个研究的猜想尼切（1962年）。这是迈向n维模量理论的试探性的第一步，这个主题有望发展成为更高维的Teichmuller理论的一个非常连贯的类似物。该项目的特点是新的概念（如自由拉格朗日）和具有挑战性的问题，一些已经准备好答案，其他更具投机性，长期性。通过精心设计，该项目将鼓励向更广泛的受众传播现代几何函数理论，这是一个近年来经历了巨大变革的经典领域，并将加强对该主题在应用中的普遍存在的理解。该项目反映了首席研究员的持续努力，通过有效的培训，创建教育材料，以及在美国和海外培养科学伙伴关系，为研究生和年轻学者提供积极和热情的研究环境。迄今为止，这一奋进对数学史上代表性不足的群体特别有效，尤其是女性。所提出的问题将发现相当大的兴趣几何分析师，特别是研究人员在这些物理相关领域的偏微分方程，变分法，和非线性弹性理论。

In this project the principal investigator initiates a study of extremal problems for mappings of finite (usually unbounded) distortion, hoping to provide background for further developments in geometric function theory and nonlinear elliptic partial differential equations. The plan of attack relies on geometric and physical intuition, drawn especially from the theory of nonlinear elasticity and materials science. This represents a trend that in recent years has become more pronounced, and it has led to increasing efforts by pure and applied mathematicians to combine such ideas and results. The project centers on the remarkable relationship (first envisioned by the principal investigator and Jani Onninen) between mappings of finite distortion and hyperelasticity. Both theories are governed by variational principles and problems of compelling common mathematical interest. The new fronts that have been created in geometric function theory include polyconvex integrals of the distortion function and the associated total n-harmonic energy. Of special interest are mappings of smallest mean distortion, whose existence, regularity, and global invertibility are related to deep unsolved mathematical questions. For instance, the present proposal takes on difficult questions concerning deformations of spherical rings in Euclidean n-space, and a much studied conjecture of Nitsche (1962). This is a tentative first step toward an n-dimensional theory of moduli, a subject that hopefully will develop into a very coherent analogue of Teichmuller theory in higher dimensions. The project features new concepts (such as free Lagrangians) and challenging questions galore, some already prepared for answers, others of a more speculative, long-term character. With its elaborate design, this project will encourage the dissemination of modern geometric function theory, a classical field that has undergone a tremendous transformation in recent years, to a wider audience and will enhance the understanding of the subject's pervasive presence in applications. The project reflects continued efforts by the principal investigator to provide an active and welcoming research environment for graduate students and young scholars through effective training, the creation of educational materials, and the fostering of scientific partnerships both within the U.S. and overseas. This endeavor has hitherto been particularly effective for groups historically underrepresented in mathematics, especially women. The proposed problems will find considerable interest among geometric analysts, especially researchers in such physically relevant fields as partial differential equations, the calculus of variations, and nonlinear elasticity theory.