Mathematical Sciences: Harmonic Analysis and Partial Differential Equations

该项目继续进行数学研究，发展了调和分析的各个方面，以期应用于线性和非线性偏微分方程式中的问题。主要的重点将是研究具有粗糙边界的区域上的椭圆边值问题。更准确地说，边界不需要具有连续的旋转切线。它们可能有角，但没有尖端。这些实际上是人们在物理表示中可以施加的最小条件，其中不精确的测量总是导致光滑度的损失。关于任何维的齐次方程的结果都被认为是很好的。不同性质的问题也不是这样。事实上，在二维中存在一个狄利克莱问题，对于该问题，比较梯度和边值的基本估计是不成立的。现在将对相应的非齐次Neumann问题进行研究，看看是否可以在适当的Soblev空间中获得函数的估计。第二条线，更几何的，涉及到用S关联的拉普拉斯度量的谱来估计黎曼度量的问题。特别是，将在不限制流形的共形类的情况下，根据其谱数据来估计三维流形的度量。这将通过构造一个其二次范数支配黎曼测度的函数来实现。这种构造需要解决流形上的一个特定的微分不等式。偏微分方程式是物理科学中数学建模的基础。人们知道，涉及运动、材料和能量等连续变化的现象遵循某些一般规律，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描述关系，而是从它们中提取定性和定量的意义。

This project continues mathematical research leading to the development of various aspects of harmonic analysis with a view toward applications to problems in linear and nonlinear partial differntial equations. The main emphasis will be on the study of elliptic boundary value problems on domains with rough boundaries. More precisely, the boundaries are not required to have a continuously turning tangent. They may have corners but not cusps. These are effectively the minimal conditions one might impose in physical representations where imprecise measurements always lead to a loss of smoothness. Results on homogeneous equations in any dimension are considered well developed. Not so are the inhomogeneous problems. In fact, in two dimensions there is a Dirichlet problem for which the fundamental estimate comparing the gradient with the boundary values does not hold. Work will now be done on the corresponding inhomogeneous Neumann problem to see if the estimates can be obtained for functions in appropriate Sobolev spaces. A second line of research, more geometric, concerns the question of estimating a Riemannian metric in terms of the spectrum of it s associated Laplacian. In particular, work will be done in estimating the metric of a three-manifold in terms of its spectral data with no restriction on the manifold's conformal class. This will be done by constructing a function whose quadratic norm dominates the Riemannian measure. The construction requires the solution of a specific differential inequality on the manifold. 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.