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

DMS 0400718题目：L无穷大中的变分，卡诺群中的完全非线性次椭圆型方程，双调和与调和映射的分析PI：王昌友，肯塔基大学。在第一部分中，PI计划继续研究L无穷大中变分计算的分析问题，包括L无限泛函的绝对无穷子与其完全非线性退化的Aronsson-Euler方程的粘性解之间的关系，以及Aronsson-Euler方程的粘性的唯一性。第二部分，基于他最近关于Carno群上的次椭圆无穷拉普拉斯方程粘性解的唯一性的工作，建立了Carnot-Caratheodory空间上包括次椭圆iSaaS-Bellman方程和水平Hessian方程的完全非线性次椭圆偏微分方程解的比较原理。在第三部分，PI计划进一步研究平稳双调和映射的部分正则性，如双调和映射奇异集的最优大小和可能结构以及四维双调和映射的热流及其应用。在第四部分，PI计划进一步研究定常双调和映射的部分正则性PI计划继续他对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的研究。所提出的问题存在于非线性偏微分方程组领域，它提供了基本的规律，并在从分析、几何、应用科学的研究问题中起着至关重要的作用。超范数变分问题不仅具有重要的数学意义，而且具有很大的实用价值。在控制机制、运筹学中的风险管理、极值工程中，人们必须为最坏的情况进行设计(例如，确定一个控制以最小化一个函数的最大值的成本泛函)，存在许多问题。另一方面，由于上确界正规函数缺乏很强的可微性，并且与其相关的偏微分方程退化为完全非线性方程，因此必须发展许多新的研究方法。许多物理现象的基础是最小能量原理，即某些构型或几何形状是通过比竞争对象具有更少的能量或面积来区分的。非线性目标约束经常导致奇点。例如，磁化材料中的磁畴壁，各种液晶材料中的点、曲线和表面缺陷，以及超导中的涡旋。我们需要发展新的数学结构和理论来解释和预测这种现象。调和映射和双调和映射的研究正是出于这些考虑。这些方向的研究成果对于我们了解具有边界非线性的二阶(或高阶)非线性椭圆型方程组以及许多潜在的应用都是非常重要的。

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.