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

点击翻译按钮获取中文摘要

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.