Sobolev Mappings of Smallest Energy

本研究计划的主题是发展函数的几何理论和弹性理论中的最小化技术。 这两个理论是通过在变分法和非线性偏微分方程的挑战性问题。 它们都依赖于几何直觉和对能量的深入分析，以最大限度地减少物质体的变形，或者更数学地说，更高维的曲面。 在寻找超弹性的数学模型时，人们必须接受并探索弹性变形的极限。 这种方法在平板和薄膜的二维理论中特别有效。 对裂纹和碎片引起的物体失效的理论预测是一个很好的动机，应该吸引纯数学家和应用领域的研究人员。 对实际问题的实验回答将导致对本项目中研究的数学问题的解决方案或更深入的见解;例如，最小平均变形的变形的存在性和唯一性。研究起源于黎曼映射定理，共形映射是Cauchy-Riemann系统的单叶解。 转移到二阶变分方程和他们的同胚解提供了新的挑战。 我们的目标是表征能量泛函的极小存在，并密切遵循共形映射。 在非线性弹性力学的数学模型中，要证明存在能量最小的变形，而这种变形又符合物质不相互渗透的原理，这是一个共同的难题。 为了建立一个可行的理论，PI将采用单调Sobolev映射作为二维弹性的合法变形。 Sobolev同胚中的极小化使得能量极小映射的存在性、唯一性和正则性等基本问题成为具有挑战性的问题。 构建新的工具来解决这些问题是这个项目的核心部分。

The main theme of this research program is to develop minimizing techniques in the geometric theory of functions and in the theory of elasticity. These two theories are related through challenging problems in the calculus of variations and nonlinear partial differential equations. They both rely on geometric intuition and an in-depth analysis of energy minimizing deformations of material bodies or, more mathematically, higher dimensional curved surfaces. In the search for mathematical models of hyperelasticity, one must accept and explore the limits of elastic deformations. This approach turns out to be particularly effective in the two-dimensional theory of flat plates and thin films. Theoretical prediction of failures of bodies caused by cracks and fractions is a good motivation that should appeal to both pure mathematicians and researchers in applied fields. Experimental answers to practical problems will lead to solutions or deeper insights into mathematical problems investigated in this project; for example, the existence and uniqueness of deformations of smallest average distortion. The research originated from the Riemann Mapping Theorem; conformal mappings being univalent solutions of the Cauchy-Riemann system. Moving to the second order variational equations and their homeomorphic solutions offers new challenges. The goal is to characterize energy-functionals whose minimizers exist and closely follow conformal maps. It is a common struggle in mathematical models of nonlinear elasticity to establish the existence of energy-minimal deformations which comply with the principle of no interpenetration of matter. To build a viable theory the PI will adopt monotone Sobolev mappings as legitimate deformations in 2-dimensional elasticity. Minimizing among Sobolev homeomorphisms the basic questions such as existence, uniqueness and regularity of energy-minimal mappings become challenging problems. Building new tools to solve such problems is the central part of this project.