Variational Approach to Geometric Function Theory

该项目的动机是在变分方法的几何函数理论（GFT）和非线性弹性（NE）的最新进展。提出的几何性质的问题起源于黎曼映射定理;共形映射是柯西-黎曼系统的单叶解。转向二阶变分方程及其同胚解提出了新的挑战。我们的目标是表征能量泛函的极小存在和类似的共形映射。在我们的极值问题中，映射在边界上是自由的。在这样的牵引自由问题中，我们关心的是能量极小解的存在性和整体可逆性。人们需要联合收割机，发展分析和拓扑学的思想。有越来越多的文献内变分方程是弱于经典的欧拉-拉格朗日方程。即使在Dirichlet积分的基本情况下，也有许多新的令人惊讶的现象。GFT目前是一个非常活跃的领域，NE的一般框架是非常富有成果和意义的。拟议的变分方法有助于这种相互作用，并导致协调一致的努力，纯数学家和应用数学家一起工作。研究课题和已经取得的成果对弹性变形、材料科学、连续介质力学等学科的发展具有潜在的影响。薄弹性膜往往呈现最小曲面的形状。问题是确定拉伸下管状薄膜的最佳形状。这涉及到调和映射的拟议研究。在模拟细胞结构、泡沫物理和组织方面似乎还有其他应用。PI将继续接待各种访问学者团体进行相互研究。

The project is motivated by recent advances in the variational approach to Geometric Function Theory (GFT) and Nonlinear Elasticity (NE). The proposed problems of geometric nature 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 resemble conformal maps. In our extremal problems mappings are free on the boundary. In such traction free problems we are concerned with existence and global invertibility of energy-minimal solutions. One needs to combine and develop the ideas of analysis and topology. There is a growing literature on inner variational equations that are weaker than the classical Euler-Lagrange equation. Even in the basic case of the Dirichlet integral there are many new surprising phenomena.GFT is currently a field of enormous activity where the general framework of NE is extremely fruitful and significant. The proposed variational approach contributes to this interplay and leads to concerted efforts of pure and applied mathematicians to work together. The research topics, and results already in place, have the potential impact on the development of elastic deformations, material science, continuum mechanics, etc. A thin elastic film tends to assume the shape of a minimal surface. The issue is to identify the optimal shape of a tubular thin film under stretch. This relates to the proposed studies of harmonic maps. There appear to be other applications in modeling cellular structures, foam physics and tissues as well. The PI will continue to host diverse groups of visiting scholars for mutual research.