Energy-Minimal Principles in Geometric Function Theory

本计画探讨能量最小化变形的几何与分析，并应用于非线性弹性的研究。后一个领域研究物理材料和物体在应力和应变下的变形，这一领域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的弹性体的一部分压缩到零体积的空间中的物理不可能性。从数学的角度来看，当试图克服物质不互穿的物理原理所假设的注入性的缺乏时，出现了严重的困难。虽然同胚解是理想模型，但与遵守上述物理原理的必要性相关的数学约束自然会导致这样的结论，即同胚解的极限必须被允许作为合法的竞争者。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得到进一步和更详细的开发。该项目还将为早期职业数学家提供研究机会，包括研究生。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者。在二维空间中，弱极限和强极限重合，并刻画了单调Sobolev映射类。非内射能量最小解是单调的，可以挤压二维平板或薄膜，但不能折叠它们。严重的挑战出现在调查中的能量最小的解决方案完全基于内变分方程。这个项目主要关注的问题类似的精神黎曼共形映射问题。这种变分问题也导致相关的索伯列夫映射问题，并在规定的边界值的情况下，需要初步调查索伯列夫变种的约旦-Schoenflies定理。该项目的另一个目标是加深几何函数理论与物理学和工程学相关领域之间的联系，促进这些不同学科从业者之间的思想交流。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。

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.