Variational approach to Geometric Function Theorem, Nonlinear PDEs and Hyperelasticy

应用科学在制定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经常通过数学论证来解释和证实实验结果，以寻求理论和实践的改进。该项目涉及偏微分方程，几何函数理论，变分法和现实世界中出现的相关问题，包括：非线性弹性，材料的微观结构和晶体仅举几例。所提出的数学对象的物理解释，如被视为弹性变形的索伯列夫同胚，已被证明有助于理解和解决非线性弹性中的挑战性问题。这里的主要目标是开发新的，并改进旧的方法，以应对这些挑战。新的数学概念，如自由拉格朗日允许建立牵引自由能最小变形的存在。霍普夫-拉普拉斯微分的进一步发展可能导致对变形中裂纹形成的预测，并有助于理解物质相互渗透的原理。裂纹引起的弹性体破坏的理论预测将对数学分析和工程领域的研究人员产生广泛的影响。在这个项目中的研究课题，虽然在一般数学上具有挑战性，也采取了适合研究生的问题。最终目标是吸引研究生和青年学者，无论男女，几何函数理论与广泛的应用，鼓励他们参与跨学科的努力，并准备和帮助他们发展有意义的互动与物理学家和工程师。首席研究员目前正在与Jani Onninen合作，为研究人员和研究生设计了一本专着，在几何函数理论。首席研究员有一个强大的努力，从纯数学和应用数学之间的相互作用中获益的历史。它导致了几个数学问题的解决，如极小曲面理论中的Nitsche猜想，关于Sobolev映射的近似的Evans-Ball猜想，变分法中自由拉格朗日的新概念，沿着Hopf微分沿着轨道的裂缝的形成，对物质相互渗透现象的严格描述，和部分答案的传说中的Morrey的猜想在二维拟凸性的秩一凸泛函，特别是，尖锐的不等式Burkholder的随机积分。后者包括复Beurling-Ahlfors奇异积分变换。这个变换的隆起之处在于它连接了复平面上一阶椭圆方程组的两个同伦类：一个是方向保持映射的同伦类，另一个是方向反转映射的同伦类。众所周知的困难问题是确定Beurling-Ahlfos变换的Lp范数。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值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.