Nonlinear PDEs in Geometric Analysis

AbstractAward：DMS-0072328首席研究员：陈文雄陈文雄教授将研究微分几何中的一系列非线性问题。他的主要课题之一是高斯曲率和标量曲率的定义：给定黎曼流形上的一个函数，它能被实现为某个点态共形度量的曲率吗？这等价于求解某些半线性椭圆型偏微分方程。 他还将致力于奇异曲面上的一致化定理和完备非紧流形之间的调和映射的存在性，这与四阶非线性椭圆方程，非线性椭圆系统和非线性抛物系统的研究密切相关。根据爱因斯坦的说法，我们生活的宇宙是一个弯曲的空间，其中重力被实现为大质量物体附近时空的扭曲或弯曲，这种形状的变化由曲率张量测量。 Chen的研究集中在理解给定函数何时可以成为曲率。这是黎曼几何学和偏微分方程分析中的一个具有挑战性的问题，许多研究人员对此做出了贡献。 本项目所研究的非线性偏微分方程和方程组在物理、化学、生物等领域有着广泛的应用，如流体动力学、燃烧理论、河流污染等

AbstractAward: DMS-0072328Principal Investigator: Wenxiong ChenProfessor Wenxiong Chen will work on a series of nonlinearproblems in differential geometry. One of his main subjects isprescribing Gaussian and scalar curvature: given a function on aRiemannian manifold, can it be realized as the curvature of somepointwise conformal metric? This is equivalent to solving certainsemi-linear elliptic partial differential equations. He willalso work on uniformization theorems on singular surfaces and theexistence of harmonic maps between complete, noncompactmanifolds, which are closely related to the study of fourth ordernonlinear elliptic equations, nonlinear elliptic systems, andnonlinear parabolic systems.According to Einstein, the Universe we live in is a curved space,in which gravity is realized as a distortion or bending ofspace-time in the neighborhood of a massive object and thischange of shape is measured by the curvature tensor. Chen'sresearch is focused on understanding when a given function canbecome a curvature. This is a challenging problem in Riemanniangeometry and the analysis of partial differential equations, towhich many researchers have contributed. The nonlinear partialdifferential equations and systems studied in this project havevarious applications in physics, chemistry, and biology, such asin fluid dynamics, combustion theory, and river pollution