Mathematical Sciences: On Nonlinear Elliptic Equations

9401815李该奖项支持在三个相关领域的几何问题的数学研究，这些研究都是基于求解各种类型的偏微分方程。第一部分讨论了n球上的标量曲率在临界点上的某些假设下的确定问题。这个问题已经在二维和三维中解决了，这个项目将考虑，在临界点上的平坦性假设，对于给定的无限可微函数，规定曲率的存在。第二个领域涉及非负高斯曲率的Weyl问题。粗略地说，这个问题关注的是一个具有给定黎曼度规和处处正高斯曲率的球体能否以标准平面度规等距嵌入三维欧几里德空间。虽然这个问题已经解决了，但对于曲率仅为非负的情况，只存在部分结果。在高斯曲率仅为非负的情况下，求出平滑等距嵌入的充分必要条件。最后，对均匀旋转白矮星的可压缩流体模型进行了研究。建立了小角速度下的平衡解，并证明了大角速度下不存在平衡解。现在必须做的是确定平衡解的全局分支的存在性——这相当于解决某个自由边界问题。偏微分方程是物理世界数学建模的基础。数学分析的作用与其说是创建方程，不如说是提供关于解的定性和定量信息。这可能包括对存在、独特性、平滑性和成长等问题的回答。此外，分析常常发展出解的近似方法和对这些近似精度的估计。＊＊＊

9401815 Li This award supports mathematical research on geometric problems in three related areas all based on solving various types of partial differential equations. The first concerns the problems of prescribing scalar curvature on the n-sphere with certain hypothesis on the critical points. The problem has been solved in dimensions two and three, this project will consider, with flatness assumptions on the critical points, the existence of a prescribed curvature(s) for a given infinitely differentiable function. The second area involves the Weyl problem with nonnegative Gauss curvature. Roughly this problem focuses on whether or not a sphere, with a given Riemannian metric an everywhere positive Gauss curvature can be isometrically embedded in three-dimensional Euclidean space with the standard flat metric. While this problem has been solved, only partial results exist for the case where the curvature is only nonnegative. To find necessary and sufficient conditions for smooth isometric embedding when the Gauss curvature is only nonnegative is the goal of this effort. Finally, work will be done in the study of the compressible fluid model of uniformly rotating white dwarf stars. Equilibrium solutions have been established for small angular velocity and the nonexistence of equilibrium solutions for large angular velocity has also been shown. What now must be done is to establish the existence of a global branch of equilibrium solutions - which amounts to solving a certain free boundary problem. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about existence, uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***