Fully Nonlinear Partial Differential Equations in Geometry

摘要奖：DMS-0505632主要研究人员：柏观这个项目的主要目的是了解一些由微分几何问题引起的非线性重椭圆型方程和抛物型方程。主要研究三类几何问题：欧氏空间和更一般流形中超曲面的平台型问题；寻找给定Weingarten曲率的闭凸超曲面的Minkowsk型问题；以及等距嵌入问题。这些问题中的每一个都涉及到一些完全非线性的偏微分方程的求解。由于它们起源于几何学，这些方程密切相关，因此有许多共同的技术问题。因此，其中一个问题的进展很可能会导致其他问题的进展，更广泛地说，在(流形上的)非线性偏微分方程组理论和几何分析方面也会取得进展。完全的非线性方程也会出现在测量学和物理学以及科学和工程的其他问题中。我们的研究结果可能会在物理学中的弦理论和最优控制理论等领域产生影响和应用。

AbstractAward: DMS-0505632Principal Investigator: Bo GuanThe primary goal of this project is to understand some nonlinearelliptic and parabolic equations which arise from problems indifferential geometry. The research will be focused on threetypes of geometric problems: Plateau-type problems forhypersurfaces in Euclidean space and in more general manifolds;the Minkowski-type problems of finding closed convexhypersurfaces of prescribed Weingarten curvature, and problems inisometric embedding. Each of these problems involves solvingsome fully nonlinear partial differential equations. Because oftheir origins in geometry, these equations are closely relatedand therefore share many common technical issues. Thus, progressin one of the problems will very likely lead to progress in theothers, and more generally in the theory of nonlinear partialdifferential equations (on manifolds) and geometric analysis.Fully nonlinear equations also arise in other problems ingeometry and physics as well as in science andengineering. Results from our research may have impact andapplications in areas such as string theory in physics, andoptimal control theory.