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

这个项目的重点是分析解偏微分方程，系统以及标量方程。我们感兴趣的方程出现在数学物理和微分几何中。重点将放在研究完全黎曼流形中的Yamabe问题，这是一个相当有趣的问题，可以用半线性椭圆方程来表示。那么问题就是找到方程的正解并分析它们的渐近行为。这项工作以一种自然的方式激发了人们通过索博列夫商的临界点来理解更大类解的努力。在用微分方程中的非线性项精确估计Hausdorff维数的集合的补中，认为解是无穷可微的。我们也会在调和映射上做一些工作，特别是研究调和映射最小化成奇异对象（如变形）的规律性。最近对两个紧黎曼流形之间调和映射的最小化正则性的一个新证明表明，在李普希茨图的目标域上展开是可能的。调和映射的正则集是人们关注的焦点。但是现在也有一些关于奇异集的一般结果。它们可能具有合理的结构，例如曲线或光滑曲线的结合。进一步的工作将确定哪些奇异集分类是可能的。偏微分方程是物理科学中数学建模的基础。涉及连续变化的现象，例如在运动、材料和能量中看到的现象，都遵循某些一般规律，这些规律可以用偏导数之间的相互作用和关系来表示。数学的关键作用不是说明关系，而是从中提取定性和定量的意义，并验证表达解决方案的方法。

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.