Fully nonlinear elliptic equations in geometry

微分方程在数学及其应用中发挥着核心作用，例如，在微分几何和数学物理中。完全非线性椭圆型和抛物型方程是目前研究人员研究的主要方程之一。在最近进展的基础上，在这个项目中，主要研究人员将建立这些方程在实或复流形上的一般存在性和正则性结果，并开发可应用于求解其他类型方程的技术工具。这个项目的结果将在几何学中产生有趣的应用，并有助于理解一般的偏微分方程。求解完全非线性椭圆和抛物型方程的一个中心问题是建立先验估计，由于方程的高度非线性和流形的一般几何以及它们的边界，先验估计带来了巨大的挑战。该项目的目标是寻找新的方法来克服这些困难。特别是，它将确定一个人可以在什么条件下求解一般流形上的方程。首席研究人员还计划研究一类结果鲜为人知的非常一般的方程，并解决复杂几何中的一些困难问题。

Differential equations play a central role in mathematics and its applications, for example, in differential geometry and mathematical physics. Fully nonlinear elliptic and parabolic equations are some of the main equations under current investigations by researchers. Building on the recent progress, in this project the principal investigator will establish general existence and regularity results for these equations on real or complex manifolds and develop technical tools which can be applied to solve other types of equations. Results in this project will yield interesting applications in geometry and help to understand general partial differential equations.A central issue in solving fully nonlinear elliptic and parabolic equations is to establish a priori estimates which present great challenges due to the high nonlinearity of the equations and the general geometry of the manifolds as well as their boundaries. The project's goal is to search for new methods to overcome these difficulties. In particular, it will identify the optimal conditions under which one can solve the equations on general manifolds. The principal investigator also plans to study a very general class of equations on which few results are known, and attack some difficult problems in complex geometry.