Variational approach to Geometric Function Theorem, Nonlinear PDEs and Hyperelasticy

几何函数定理、非线性偏微分方程和超弹性的变分法

基本信息

批准号：
1802107
负责人：
Tadeusz Iwaniec
金额：
$ 27万
依托单位：
Syracuse University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-06-01 至 2022-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1802107&HistoricalAwards=false
关键词：
Variational approach Geometric Function Theorem

项目摘要

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.

应用科学对于制定和解决有趣的数学问题很重要。相反，科学和工程领域的研究人员通常通过数学论点寻求改进理论和实践的改进，以解释和确认实验结果。该项目涉及部分微分方程，几何函数理论，变异的计算以及在现实世界应用中出现的相关问题，包括：非线性弹性，材料的微结构和晶体，仅举几例。事实证明，所提出的数学对象的物理解释，例如被视为弹性变形的Sobolev同构，已被证明可用于理解和解决非线性弹性中的具有挑战性的问题。这里的主要目的是开发新的，并改善旧方法来应对这些挑战。新的数学概念（例如自由拉格朗日）允许人们建立牵引力自由能量最小变形的存在。 Hopf-Laplace差异的进一步进步可能会导致对变形中裂纹的形成以及理解物质互穿原理的预测。裂纹引起的弹性物体失败的理论预测将对工程领域的数学分析师和研究人员产生广泛的影响。该项目的研究主题虽然在数学上一般具有挑战性，但也提出了适合研究生的问题。最终的目标是吸引研究生和年轻学者，包括男女，以广泛的应用来吸引几何功能理论，以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。这位主要研究人员目前正在与Jani Onninen合作，为研究人员以及几何功能理论的研究生设计的联合专着。它导致了几个数学问题的解决方案，例如在最小表面理论中的nitche猜想，埃文斯·波尔的猜想与sobolev mapping ting ting sobolev sappings ting ting-nek groundofomorists的近似，自由lagrangians的新颖概念，在变化的计算中，沿料理的轨迹形成的裂纹界限和形成的型号的形成，是料理型的形成，并描述了严格的描述。维度二中的部分答案是传奇的莫雷（Morrey）对等级孔函数的Quasiconvexity的猜想；尤其是伯克持有人随机积分的急剧不平等。后者包括复杂的beurling-ahlfors单数积分变换。这种转换的杰出性在于它在复杂平面中连接了一阶椭圆系统的两个同型类别：一个其解决方案是定向映射的，另一个是在方向上逆转解决方案。众所周知的困难问题是识别beurling-ahlfos变换的LP-norm。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的审查标准来评估值得支持的。