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

提案DMS 0400718题目：L-无穷大变分法，卡诺群中的完全非线性次椭圆方程，双调和映射和调和映射的分析PI：王长友，剑桥大学摘要该提案包括四个主要领域的问题。在第一部分中，PI计划继续研究L-无穷远变分计算的分析问题，包括L-无穷远泛函的绝对极小与其完全非线性退化方程的Aronsson-Euler方程的粘性解之间的关系，以及Aronsson-Euler方程粘性的唯一性。在第二部分中，基于他最近关于Carnot群上的亚椭圆无穷Laplace方程粘性解的唯一性的工作，我们建立了Carnot-Caratheodory空间上的完全非线性亚椭圆方程，包括亚椭圆Isaas-Bellman方程和水平Hessian方程的比较原理。PI计划进一步研究平稳双调和映射的部分正则性，例如双调和映射奇异集的最佳大小和可能结构以及热在第四部分中，PI计划继续研究调和映射的弱序列紧性和能量量子化以及调和映射的热流，所提出的问题属于非线性偏微分方程领域，它为研究分析、几何、应用科学等问题提供了基本规律，起着至关重要的作用。超范数变分问题不仅在数学上很重要，而且具有很大的实际意义。在控制机制、运筹学中的风险管理、极端价值工程等方面存在许多问题，其中必须针对最坏情况进行设计（例如，确定控制以最小化成本泛函，即函数的最大值）。另一方面，由于上确界正规泛函缺乏强可微性，且其所对应的偏微分方程是退化的完全非线性方程，因此需要发展许多新的研究方法。许多物理现象的基础是一个最小能量原理，在这个原理中，某些构型或几何形状的能量或面积比竞争对象小，非线性目标约束常常导致奇异性。例如，磁化材料中的畴壁，各种液晶材料中的点、曲线和表面缺陷，以及超导性中的涡旋。我们需要发展新的数学结构和理论来解释和预测这些现象。调和映射和双调和映射的研究正是出于这些考虑。这些方面的研究成果对我们认识二阶（或更高阶）边界非线性椭圆型方程组及其许多潜在的应用具有重要意义。

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.