Mathematical Sciences: Harmonic Analysis and Partial Differential Equations

该项目继续进行数学研究，以促进谐波分析的各个方面的发展，以期应用于线性和非线性偏微分方程的问题。重点是研究粗糙边界域上的椭圆边值问题。更准确地说，边界不需要有连续的转弯切线。它们可能有角，但没有尖。这些实际上是在物理表示中可能施加的最小条件，其中不精确的测量总是导致平滑性的损失。在任何维度上齐次方程的结果被认为是很成熟的。非齐次问题则不是这样。事实上，在二维中存在一个Dirichlet问题，其中比较梯度与边界值的基本估计不成立。现在将研究相应的非齐次Neumann问题，看看是否可以在适当的Sobolev空间中获得函数的估计。研究的第二条线，更几何化，关注黎曼度规的相关拉普拉斯谱的估计问题。特别是，我们将在不受流形保形类限制的情况下，根据其光谱数据估计三流形的度量。这将通过构造一个二次范数支配黎曼测度的函数来实现。这种构造要求流形上一个特定微分不等式的解。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。

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.