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

University Of Kentucky Research Foundation, Lexington KY

Investigators

Abstract

Proposal DMS 0400718 Title: Calculus of variations in L-infinity, fully nonlinear subelliptic equations in Carnot groups, analysis of biharmonic and harmonic maps PI: Changyou Wang, University of Kentucky ABSTRACT The proposal consists of problems in four main areas. In part 1, the PI plans to continue his study on analytic issues of calculus of variations in L-infinity consisting of the relation between absolute minimizers of L-infinity functionals and viscosity solutions of their Aronsson-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 to the subelliptic infinity-Laplace equation on Carnot groups, the PI aims to establish the comparison principle for fully nonlinear subelliptic pdes including subelliptic Isaas-Bellman equations and horizontal Hessian equations on Carnot-Caratheodory spaces. In part 3, the PI plans to further study partial regularity of stationary biharmonic maps such as the optimal size and possible structures of the singular set of biharmonic maps and the heat flow of biharmonic maps in four dimension and its applications. In part 4, the PI plans to continue his study on weak sequential compactness and energy quantization of harmonic maps and the heat flows of harmonic maps. The proposed problems lie in the field of nonlinear pdes which provide basic laws and play crucial roles in studying problems from analysis, geometry, applied sciences. Variational problems of supernorm are not only mathematically important but also of great practical interests. There are many problems from control mechanisms, risk managements in operation research, extreme value engineering where one must design for the worst case (e.g. determine a control to minimize the cost functional which is the maximum of a function). On the other hand, since supremum norm functionals lack strong differentiability and their associated pdes are degenerate fully nonlinear equations, many new techniques must be developed for the study. Underlying many physical phenomena is a least energy principle where certain configurations or geometric shape are distinguished 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 surface defects in various liquid crystal materials, and vortices in superconductivity. We need to develop new mathematical structures and theories in order to explain and predict such phenomena. The proposed study on harmonic maps and biharmonic maps is certainly motivated by these considerations. The research findings in these directions shall be very important to our knowledge of second (or higher) order nonlinear elliptic systems with borderline nonlinearities and many potential applications as well.