Mathematical Sciences: Existence, Singularity & Regularity of Solutions of Scalar Equations & Systems in Connection with Differential Geometry & Mathematical Physic

本项目的重点是分析解决方案 偏微分方程、系统以及标量 方程 有趣的方程出现在数学中 物理学和微分几何 重点将放在 完备黎曼流形上Yamabe问题的研究 流形，一个相当感兴趣的问题，可以是 用半线性椭圆方程表示。 的 问题是找到一个积极的解决办法， 方程并分析其渐近行为。 这项工作 以自然的方式激励人们努力理解更大的类 的解决方案，通过临界点的索伯列夫商。 解被认为是无穷可微的， Hausdorff维数恰好为 通过微分中的非线性项估计 方程 工作也将在调和映射，特别是 在研究极小化调和映射到 奇异的物体，比如可变折叠。最近的一个新的证明， 两个紧映射之间极小调和映射的正则性 黎曼流形表明，展开是可能的， 目标域是Lipschitz图。多数人的注意力 对调和映射的正则集进行了研究。 但 现在有一些关于奇异集的一般结果， 好. 它们可能具有合理的结构，例如曲线或 平滑曲线的联合。 将开展进一步工作，以确定 奇异集合的分类是可能的。 偏微分方程是 物理科学中的数学建模。 的现象 包括连续变化，如运动、材料 和能量都服从某些普遍规律， 可以用相互作用和关系来表达 偏导数之间的关系 数学的关键作用不是 来陈述这些关系，而是为了提取定性的 和量化的意义，并验证方法， 表达解决方案。

The focus of this project is the analysis of solution of partial differential equations, systems as well as scalar equations. The equations of interest arise in mathematical physics and differential geometry. Emphasis will be placed on the study of the Yamabe problem in a complete Riemannian manifold, a problem of considerable interest which can be formulated in terms of semi-linear elliptic equations. The problem is then one of finding positive solutions to the equations and analyzing their asymptotic behavior. This work motivates in a natural way efforts to understand a larger class of solutions by means of critical points of the Sobolev quotient. The solutions are believed to be infinitely differentiable in the complement of sets whose Hausdorff dimension is precisely estimated by means of the nonlinear term in the differential equation. Work will also be done on harmonic maps, particularly in studying the regularity of minimizing harmonic maps into singular objects like varifolds. A recent new proof of the regularity of minimizing harmonic maps between two compact Riemannian manifolds suggests that an expansion is possible to target domains which are Lipschitz graphs. Most of the attention is paid to the sets of regularity for harmonic maps. But there are now some general results concerning the singular sets as well. They may have reasonable structures, such as curves or unions of smooth curves. Further work will be done to determine what classifications of singular sets are possible. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them and validate methods for expressing solutions.