Variational Approach to Geometric Function Theory

几何函数理论的变分法

• 批准号: 1301570
• 负责人: Jani Onninen
• 金额: $ 15.12万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1301570&HistoricalAwards=false
• 关键词: Variational; Approach; Geometric; Function; Theory

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.

该项目受到几何函数理论（GFT）和非线性弹性（NE）变分方法的最新进展的推动。提出的几何性质问题源于黎曼映射定理；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。目标是表征其最小值存在且类似于保角映射的能量泛函。在极值问题中，边界上的映射是自由的。在这类无牵引问题中，我们关注能量极小解的存在性和全局可逆性。人们需要结合和发展分析和拓扑的思想。越来越多的文献研究了比经典欧拉-拉格朗日方程更弱的内变分方程。即使在狄利克雷积分的基本情况下，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，其中NE的总体框架非常富有成果和意义。所提出的变分方法有助于这种相互作用，并导致纯数学家和应用数学家共同努力工作。这些研究课题和已经取得的成果对弹性变形学、材料科学、连续介质力学等学科的发展具有潜在的影响。一层薄薄的弹性膜往往呈现最小表面的形状。问题是确定管状薄膜在拉伸下的最佳形状。这与提议的谐波图研究有关。在模拟细胞结构、泡沫物理和组织方面也有其他应用。PI将继续接待不同类型的访问学者，进行相互研究。