Mathematical Sciences: Monge-Ampere Type Equations and Related Problems in Differential Geometry

9626722关这个项目追求的是几何偏微分方程。在分析微分几何中的许多问题时，往往会产生与非线性偏微分方程组有关的问题。研究者计划研究Monge-Ampere型方程，重点研究退化情况下解的正则性；Christoffel-Minkowski问题和相关的Hessian方程；指定曲率的超曲面的Dirichlet问题；超曲面的曲率发展方程。与线性偏微分方程组不同的是，关于非线性偏微分方程组的一般理论很少。然而，在各种几何环境中出现的许多问题只能用非线性微分方程组来表示。这项拟议的研究代表了从几何角度探索这些方程的一种尝试--几何方法的主要优势似乎是，一个人能够做出强大的、但具有几何意义的假设，从而简化方程。

9626722 Guan This project pursues geometric partial differential equations. Analysis of many problems in differential geometry often leads to questions concerning nonlinear partial differential equations. The investigator plans to study Monge-Ampere type equations with an emphasis on the regularity of solutions in the degenerate case; the Christoffel-Minkowski problems and related Hessian equations; the Dirichlet problem for hypersurfaces of prescribed curvatures; curvature evolution equations for hypersurfaces. Unlike linear partial differential equations there is little general theory regarding nonlinear partial differential equations. Yet many problems arising in various geometric settings can only be formulated as nonlinear differential equations. And the proposed research represents an attempt to pursue these equations from a geometric perspective - the main advantage of the geometric approach seems to be that one is able to make fairy strong, but geometrically meaningful, assumptions thereby simplifying the equations.