Sobolev Mappings of Smallest Energy

本研究计划的主要主题是发展几何函数理论和弹性理论中的最小化技术。这两个理论通过变分演算和非线性偏微分方程中的挑战性问题联系在一起。它们都依赖于几何直觉和对最小化物质体变形的能量的深入分析，或者更精确地说，高维曲面。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被证明是特别有效的。对裂纹和分数引起的物体破坏进行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。对实际问题的实验答案将导致本项目所研究的数学问题的解决方案或更深入的见解；例如，最小平均畸变变形的存在唯一性。该研究起源于黎曼映射定理；Cauchy-Riemann系统的保角映射是一元解。转向二阶变分方程及其同胚解提供了新的挑战。目标是表征能量泛函，其最小值存在并密切遵循保角映射。在非线性弹性数学模型中，如何建立符合物质无互穿原则的能量最小变形的存在性是一个常见的难题。为了建立一个可行的理论，PI将采用单调Sobolev映射作为二维弹性中的合法变形。在Sobolev同胚中最小化，能量最小映射的存在性、唯一性和正则性等基本问题成为具有挑战性的问题。构建解决这些问题的新工具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.