Mathematical Sciences: On Nonlinear Elliptic Equations

小行星9401815 该奖项支持三个相关领域的几何问题的数学研究，所有这些都基于解决各种类型的偏微分方程。 第一个问题是在对临界点作一定假设的情况下，给出n-球面上的标量曲率。 这个问题已经在二维和三维中得到了解决，这个项目将考虑，在临界点上的平坦性假设，给定无穷可微函数的规定曲率的存在性。 第二个领域涉及具有非负高斯曲率的Weyl问题。 粗略地说，这个问题集中在一个球面，在一个给定的黎曼度量下，一个处处正的高斯曲率是否可以等距地嵌入到具有标准平坦度量的三维欧氏空间中。 虽然这个问题已经解决，只有部分结果存在的情况下，曲率只有非负的。 当高斯曲率仅为非负时，寻找等距光滑嵌入的充要条件是本文的目标。 最后，我们还将研究匀速旋转白色矮星的可压缩流体模型。 对于小的角速度已经建立了平衡解，对于大的角速度也已经证明了平衡解的不存在。 现在必须做的是建立平衡解的整体分支的存在性--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建立方程，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、唯一、平滑和增长等问题的答案。 此外，分析经常发展出解的近似方法和对这些近似的准确性的估计。 ***

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. ***