Geometric Analysis of Deformations of Finite Distortiion via Nonlinear PDEs and Null Lagrangians

通过非线性偏微分方程和零拉格朗日量对有限畸变变形进行几何分析

• 批准号: 0301582
• 负责人: Tadeusz Iwaniec
• 金额: $ 35.93万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0301582&HistoricalAwards=false
• 关键词: Geometric; Analysis; Deformations; Finite; Distortiion

Project Abstract: DMA 0301582T Iwaniec, Syracuse University Intellectual Merit. Prominent advances in analysis will continue toenhance both theory and applications. In this challenge, there is enormousopportunity for contributions from geometric function theory andnonlinear partial differential equations. Broadly speaking, this theory isabout weakly differentiable mappings which distort small circles in a controlled manner (finite distortion). It elevates allthe analytic and geometric spirit of holomorphic functions in the complexplane to higher dimensions; domains of the Euclidean n-space orRiemannian manifolds, Mappings of finite distortion owe much of theirimportance to recent developments inseveral fields of applied mathematics: continuum mechanics, nonlinearelasticity, material science, composites with microstructure, crystals,and so forth. However, in this proposal we focus on practical questionsonly through mathematical formulations, sometimes without proclaimingdefinite answers. We emphasize the fundamental role of the Jacobiandeterminant and other nonlinear differential expressions, called nullLagrangians, which have driven (over the past twelve years) a veryproductive study of mappings with unbounded distortion. The presentknowledge of null Lagrangians will tell us something about the regularityand topological behavior of those mappings. This also brings us closer tothe methods of harmonic analysis and nonlinear partial differential equations. However, we departfrom the earlier theories, which are governed byuniformly elliptic systems, and move into the realm of degenerate ellipticequations, where important new applications lie. Broader Impact.Hopefully this proposal gives some appreciation of thediversity of applications and directions in which current research of thePI is moving, as well as a glimpse of the substantial body of joint workwith young scholars. It is at this point where the proposed research meetsthe broader impact criteria, such as: effective training and educationalmaterials, enhancing partnerships, participation of under-representedgroups from non-PhD granting institutions in the USA and abroad, women andmen. The Principal Investigator plans to organize a conference at Syracuse to bring togetherresearchers in geometric function theory and in applied areas. Theproposed program reflects continued efforts of the PI to reach outgraduate students. In the final analysis, education is the path toopportunities for countless individuals, and the mathematical community asa whole is the beneficiary.

项目摘要：DMA 0301582T Iwaniec，锡拉丘兹大学智力功勋。分析领域的显著进步将继续促进理论和应用的发展。在这一挑战中，几何函数理论和非线性偏微分方程有巨大的贡献机会。广义地说，这个理论是关于弱可微映射，它以受控的方式扭曲小圆(有限扭曲)。它将复平面上全纯函数的所有解析和几何精神提升到更高的维度；欧几里得n空间或黎曼流形的区域、有限变形映射在很大程度上归功于应用数学的几个领域的最新发展：连续介质力学、非线性相对论、材料科学、微结构复合材料、晶体等。然而，在这个提案中，我们只通过数学公式来关注实际问题，有时没有宣布明确的答案。我们强调雅可比行列式和其他称为零拉格朗日的非线性微分表达式的基本作用，在过去的12年里，它们推动了对具有无界失真的映射的非常富有成效的研究。零拉格朗日的现有知识将告诉我们一些关于这些映射的正则性和拓扑行为的信息。这也使我们更接近调和分析和非线性偏微分方程的方法。然而，我们摆脱了由一致椭圆组支配的早期理论，进入退化椭圆方程域，在那里有重要的新应用。更广泛的影响。希望这项建议能对当前对国际和平研究所的应用和研究方向的多样性给予一些赞赏，并使人们得以一窥与年轻学者的大量合作。正是在这一点上，拟议的研究符合更广泛的影响标准，如：有效的培训和教育材料，加强伙伴关系，美国和国外非博士学位授予机构代表不足的群体的参与，女性和男性。首席研究人员计划在锡拉丘兹组织一次会议，将几何函数理论和应用领域的研究人员聚集在一起。拟议的计划反映了PI为接触研究生所做的持续努力。归根结底，教育是无数人获得机会的途径，而整个数学界是受益者。