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)偏差映射的极值问题，希望为几何函数理论和非线性椭圆型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得出的直觉。这代表了一种近年来变得更加明显的趋势，这导致纯粹的和应用的数学家越来越努力地将这些想法和结果结合在一起。该项目的中心是有限变形映射和超弹性映射之间的显著关系(最初是由首席研究员和贾尼·奥尼宁设想的)。这两种理论都受到变分原理和引人注目的共同数学兴趣问题的支配。几何函数理论中产生的新前沿包括变形函数的多凸积分和相关的总n-调和能量。特别令人感兴趣的是平均偏差最小的映射，其存在性、正则性和全局可逆性与深层次的未解决的数学问题有关。例如，本方案解决了球环在欧几里得n-空间中变形的难题，以及Nitsche(1962)的一个被广泛研究的猜想。这是迈向n维模理论的试探性的第一步，这一主题有望发展成为更高维度上非常连贯的泰希穆勒理论的类比。该项目以新概念(如自由拉格朗日)和充满挑战性的问题为特色，其中一些已经为答案做好了准备，另一些则更具投机性和长期性。凭借其精心的设计，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化的经典领域，并将加强对该学科在应用中普遍存在的理解。该项目反映了首席研究人员的持续努力，通过有效的培训、教育材料的创建以及促进美国和海外的科学合作伙伴关系，为研究生和年轻学者提供一个积极和受欢迎的研究环境。到目前为止，这一努力对历来在数学领域代表性不足的群体尤其有效，尤其是女性。几何分析家，特别是偏微分方程、变分和非线性弹性理论等物理相关领域的研究人员，将对所提出的问题产生相当大的兴趣。

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.