Variational approach to Geometric Function Theorem, Nonlinear PDEs and Hyperelasticy

应用科学在表述和解决有趣的数学问题方面很重要。相反，科学和工程领域的研究人员往往通过数学论证来解释和证实实验结果，以寻求理论和实践上的改进。该项目涉及偏微分方程组、几何函数理论、变分和实际应用中出现的相关问题，包括：非线性弹性、材料的微观结构和晶体等。所提出的数学对象的物理解释，如被视为弹性变形的索博列夫同胚，已被证明在理解和解决非线性弹性中的挑战性问题方面是有用的。这里的主要目标是开发新的、改进旧的方法来应对这些挑战。自由拉格朗日等新的数学概念使人们能够证明牵引力自由能--最小变形的存在。霍普夫-拉普拉斯微分学的进一步进展可能导致对变形中裂纹形成的预测，以及对物质互穿原理的理解。弹性体裂纹体破坏的理论预测将对工程领域的数学分析人员和研究人员产生广泛的影响。这个项目的研究主题，虽然总体上在数学上具有挑战性，但也包含了适合研究生的问题。最终目标是吸引研究生和年轻学者，包括女性和男性，学习具有广泛应用的几何函数理论，鼓励他们参与跨学科的努力，并准备和帮助他们与物理学家和工程师发展有意义的互动。首席研究人员目前正在与贾尼·奥尼宁合作撰写一本专为几何函数论的研究人员和研究生设计的联合专著。主要研究人员有从纯数学和应用数学之间的相互作用中获益的强大努力的历史。它解决了极小曲面理论中的Nitsche猜想、关于具有微分同胚的Sobolev映射逼近的Evans-Ball猜想、变分中自由拉格朗日量的新概念、沿Hopf微分轨迹的裂纹的形成、物质互穿现象的严格描述、以及传奇的Morrey关于一阶凸泛函的拟凸性的猜想在二维的部分答案；特别是Burkhold随机积分的尖锐的不等式。后者包括复数Beurling-Ahlfors奇异积分变换。这种变换的突出之处在于它连接了复平面上一阶椭圆组的两个同伦类：一个的解是保向映射，另一个是方向相反的解。出了名的困难问题是确定Beurling-Ahlfos变换的Lp范数。这一问题的现状是前所未有的。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Applied sciences are important in formulating and solving interesting mathematical problems. Conversely, researchers in science and engineering fields often seek improvements in both theory and practice via mathematical arguments to explain and confirm experimental results. This project involves partial differential equations, geometric function theory, calculus of variations and related problems that arise in real-world applications including: nonlinear elasticity, microstructure of materials, and crystals to name a few. The proposed physical interpretations of mathematical objects, like Sobolev homeomorphisms that are viewed as elastic deformations, have proven useful in understanding and solving challenging problems in nonlinear elasticity. The primary aim here is to develop new, and improve old, methods to meet these challenges. The new mathematical concepts such as free Lagrangians allow one to establish the existence of traction free energy-minimal deformations. Further advances in Hopf-Laplace differentials could lead to predictions of the formation of cracks in deformations, and in understanding the principle of interpenetration of matter. Theoretical prediction of failure of elastic bodies caused by cracks would have a broad impact to both mathematical analysts and researchers in the engineering fields. The research topics in this project, although in general mathematically challenging, also take on questions that are suitable for graduate students. The ultimate goal is to attract graduate students and young scholars, both women and men, to geometric function theory with a wide range of applications, to encourage them to participate in the interdisciplinary efforts, and to prepare and help them to develop meaningful interactions with physicists and engineers. The principal investigator is currently working with Jani Onninen on a joint monograph designed for researchers as well as graduate students in geometric function theory.The principal investigator has a history of strong efforts to gain from an interplay between pure and applied mathematics. It resulted in the solution of several mathematical problems such as the Nitsche conjecture in the theory of minimal surfaces, the Evans-Ball conjecture on approximation of Sobolev mappings with diffeomorphisms, the novel concept of free Lagrangians in the calculus of variations, formation of cracks along trajectories of Hopf differentials, rigorous description of the phenomenon of interpenetration of matter, and the partial answers in the dimension two to the legendary Morrey's conjecture on quasiconvexity of the rank-one convex functionals; in particular, sharp inequalities of Burkholder's stochastic integrals. The latter include the complex Beurling-Ahlfors singular integral transform. The eminence of this transform lies in the fact that it connects two homotopy classes of the first order elliptic systems in the complex plane: one whose solutions are orientation preserving mapping and the other with orientation reversing solutions. The notoriously difficult problem has been to identify the Lp-norm of the Beurling-Ahlfos transform. The status of the problem goes on as never before.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.