Mathematical Sciences: Methods for Solutions of Some Nonlinear Eliptic Equations

陈教授将研究具有球面拓扑类型的流形上曲率的规定问题。此外，他将尝试用变分方法来证明经典的均匀化定理，并证明该定理在具有圆锥奇点的曲面上的推广。变分方法还将用于对一些微分几何的非线性偏微分方程的解进行分类，并研究欧几里德空间中域的非线性椭圆边值问题的解。该奖项支持的研究通常被称为全球分析。它试图找到并描述曲面或流形上非线性偏微分方程的全局解。这些方程描述了曲面的局部性质，如曲率。因此，这项研究力求将当地资料与全球资料联系起来。

Professor Chen will work on the problem of prescribing curvature on manifolds which have the topological type of a sphere. In addition, he will attempt to apply variational methods to reprove the classical uniformization theorem and prove an extension of the theorem to surfaces with conical singularities. Variational methods will also be used to classify solutions of some nonlinear partial differential equations of differential geometry and to study solutions of nonlinear elliptic boundary value problems for domains in Euclidean space. The research supported by this award is frequently referred to as global analysis. It attempts to find and describe global solutions to nonlinear partial differential equations on a surface or manifold. These equations describe local properties of the surface such as curvature. The research, therefore, seeks to relate local information to global information.