Energy-Minimal Principles in Geometric Function Theory

本项目探讨几何和分析的能量最小化变形与应用研究的非线性弹性。后一个领域，研究物理材料和物体在应力和应变下的变形，受到材料科学和物理学发展的影响。在非线性弹性数学模型的发展中面临的一个共同挑战是如何解释将正体积弹性体的一部分压缩到零体积空间中的物理不可能性。从数学的观点来看，当试图克服物质不相互渗透的物理原理所假定的缺乏注入性时，会出现严重的困难。虽然同胚解是理想的模型，但与遵守上述物理原理的必要性相关的数学约束自然导致同胚解的极限必须被允许作为合法的竞争对手。新的分析和几何工具已经开发出来以适应这些困难，这些工具将在本项目中进一步和更详细地开发。该项目还将为包括研究生在内的早期职业数学家提供研究机会。Sobolev同胚的能量最小化序列的弱极限是能量最小化的自然候选。在二维空间中，弱极限和强极限重合并表征了单调Sobolev映射的类别。非内射能量极小解是单调的，可以挤压二维板或薄膜，但不能折叠它们。单纯基于内变分方程的能量极小解的研究面临着严峻的挑战。这个项目主要关注与黎曼保角映射问题在精神上类似的问题。这样的变分问题也会导致相关的Sobolev映射问题，并且在规定边界值的情况下，需要对jordan - schoen苍蝇定理的Sobolev变分进行初步研究。该项目的进一步目标是通过促进这些不同学科的实践者之间的思想交流，加深几何函数理论与物理和工程相关领域之间的联系。该奖项反映了美国国家科学基金会的法定使命，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。

This project investigates the geometry and analysis of energy-minimizing deformations with applications to the study of nonlinear elasticity. The latter field, which studies the deformation of physical materials and bodies in response to stress and strain, is informed by developments in materials science and physics. A common challenge faced in the development of mathematical models of nonlinear elasticity is how to account for the physical impossibility of compressing a portion of an elastic body of positive volume into a space of zero volume. From a mathematical point of view, serious difficulties arise when trying to overcome the lack of injectivity as postulated by the physical principle of non-interpenetration of matter. While homeomorphic solutions would be ideal models, mathematical constraints linked to the necessity of complying with the aforementioned physical principle lead naturally to the conclusion that limits of homeomorphic solutions must be allowed as legitimate competitors. New analytic and geometric tools have been developed to accommodate these difficulties, and those tools will be developed further and in greater detail in this project. The project will also afford research opportunities for early career mathematicians, including graduate students.Weak limits of energy-minimizing sequences of Sobolev homeomorphisms are natural candidates of energy-minimizers. In two dimensions, weak and strong limits coincide and characterize the class of monotone Sobolev mappings. Non-injective energy-minimal solutions, being monotone, may squeeze two-dimensional plates or thin films, but may not fold them. Serious challenges arise in the investigation of energy-minimal solutions solely based on inner-variational equations. This project is largely concerned with questions similar in spirit to the Riemann conformal mapping problem. Such variational questions also lead to associated Sobolev mapping problems and, in the case of prescribed boundary values, require a preliminary investigation of Sobolev variants of the Jordan-Schoenflies theorem. A further goal of the project is to deepen the connections between geometric function theory and relevant areas within physics and engineering, by fostering the exchange of ideas among practitioners of these various subjects.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.