Calculus of Variations in L-infinity, Fully Nonlinear Subelliptic Equations on Carnot Groups, Analysis of Biharmonic Maps and Harmonic Maps

L-无穷变分微积分、卡诺群上的完全非线性次椭圆方程、双调和映射和调和映射分析

• 批准号: 0400718
• 负责人: Changyou Wang
• 金额: $ 7.23万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400718&HistoricalAwards=false
• 关键词: Calculus; Variations; infinity; Fully; Nonlinear

Proposal DMS 0400718Title: Calculus of variations in L-infinity, fully nonlinearsubelliptic equations in Carnot groups, analysis of biharmonic andharmonic mapsPI: Changyou Wang, University of KentuckyABSTRACTThe proposal consists of problems in four main areas. In part 1,the PI plans to continue his study on analytic issues of calculusof variations in L-infinity consisting of the relation between absoluteminimizers of L-infinity functionals and viscosity solutions of theirAronsson-Euler equations which are fully nonlinear degenerate pdes,and uniqueness of viscosity of Aronsson-Euler equations. In part 2,based on his recent works on uniqueness of viscosity solutions tothe subelliptic infinity-Laplace equation on Carnot groups, the PIaims to establish the comparison principle for fully nonlinearsubelliptic pdes including subelliptic Isaas-Bellmanequations and horizontal Hessian equations on Carnot-Caratheodory spaces.In part 3, the PI plans to further study partial regularity ofstationary biharmonic maps such as the optimal size and possiblestructures of the singular set of biharmonic maps and the heat flowof biharmonic maps in four dimension and its applications.In part 4, the PI plans to continue his study on weaksequential compactness and energy quantization of harmonic mapsand the heat flows of harmonic maps.The proposed problems lie in the field of nonlinear pdes whichprovide basic laws and play crucial roles in studying problemsfrom analysis, geometry, applied sciences. Variational problems ofsupernorm are not only mathematically important but also ofgreat practical interests. There are many problems from controlmechanisms, risk managements in operation research, extreme valueengineering where one must design for the worst case(e.g. determine a control to minimize the cost functional which is themaximum of a function). On the other hand, since supremum normfunctionals lack strong differentiability and their associated pdesare degenerate fully nonlinear equations, many new techniques must bedeveloped for the study. Underlying many physical phenomena is a leastenergy principle where certain configurations or geometric shape aredistinguished by having less energy or area than competing objects.The nonlinear target constraints often lead to singularities. For example,domain walls in magnetized materials, point, curve, and surfacedefects in various liquid crystal materials, and vortices insuperconductivity. We need to develop new mathematical structuresand theories in order to explain and predict such phenomena. Theproposed study on harmonic maps and biharmonic maps is certainlymotivated by these considerations. The research findings in thesedirections shall be very important to our knowledge of second (or higher)order nonlinear elliptic systems with borderline nonlinearitiesand many potential applications as well.

提案DMS 0400718TITLE：Carnot组的L侵入性变异的计算，完全非线性的临界方程，对Biharmonic Andharmonic Mapspi的分析：肯塔基亚比大学（University of Kentuckyabstract）的Biharmonic AndHarmonic Mapspi：Changyou Wang，肯塔基亚大学（University of Kentuckyabstract）的提议与四个主要地区的问题构成了问题。在第1部分中，PI计划继续他对L-内度计算变化的分析问题的研究，该问题包括Linfinity功能的绝对聚物与Houlsaronsson-Euler方程的绝对异构体之间的关系，这些方程是完全非线性的PDES，以及Aronsson-Eulonsson-Eulerererererererererererererererererererererererererererererererererererererererererererererererererererererererererererererer的独特性。 In part 2,based on his recent works on uniqueness of viscosity solutions tothe subelliptic infinity-Laplace equation on Carnot groups, the PIaims to establish the comparison principle for fully nonlinearsubelliptic pdes including subelliptic Isaas-Bellmanequations and horizo​​ntal Hessian equations on Carnot-Caratheodory spaces.In part 3, the PI plans to further study partial regularity统一的双谐图像，例如奇异的奇异图中的最佳尺寸和可能性，在四个方面及其应用中的双谐图和双谐图的热流动图。从分析，几何学，应用科学研究问题中的关键作用。 supernorm的变分问题在数学上不仅重要，而且是实践兴趣。从控制机制，操作研究中的风险管理，极值工程化中，必须在最坏情况下设计一些问题（例如，确定控制权以最大程度地减少成本功能，这是功能的最大值）。另一方面，由于最高规范功能缺乏强大的可不同性，并且其相关的PDESARE脱成为完全非线性的方程式，因此许多新技术必须进行研究。基本的许多物理现象是一种最小二能的原理，其中某些构型或几何形状与竞争对象相比，能量或面积少。非线性目标限制通常会导致奇异性。例如，各种液晶材料中的磁化材料，点，曲线和表面曲线的结构壁和涡流不良。我们需要开发新的数学结构和理论，以解释和预测这种现象。这些考虑因素肯定会激发有关谐波图和谐波图的规定研究。对于我们对第二（或更高）阶的非线性椭圆系统的了解，具有边缘性非线性和许多潜在应用也非常重要。