Multidirectional Boundry Value Problems

该方法提出求解n维欧几里德空间多面体域上的线性椭圆型微分方程和系统的边值问题。数据在边界上定义的Lebesgue空间和Sobolev空间中规定，并在点向非切极限的意义上取值。微分方程的阶不受限制。具有流形边界的多面体域给出了局部连续横向向量场不存在的例子；即边界不能局部实现为直角坐标下的函数图。在n=3这样的低维空间中也是如此。这一事实使得在边界（Rellich恒等式）上获得先验能量估计变得困难，特别是对于系统和高阶方程。当这些估计可以得到时，就可以证明数据的闭子空间的存在性。子空间是数据的整个巴拿赫空间，使用了一种连续性方法，这种方法似乎需要多面体边界的组合结构。E. E. Moise的一个定理在四维空间中提供了这种结构，但是在高维空间的例子显示了它的缺乏。PI计划寻找计算方法来克服这个问题。线性椭圆边值问题的两个标准例子是：(1)根据边界上的分布（数据）推导出固体内部的稳态温度分布，以及(2)根据边界位移或边界上的规定应力的知识推导出弹性体内部的应力、应变和位移系统。在三维空间中，自然存在多面体形式的物理对象（例如晶体结构）。事实上，我们有理由认为，这种情况比无限光滑或无限粗糙的边界表面（如利普希茨函数所描述的边界表面）发生得更频繁。多面体域可以有不能用函数图来描述的边界，这可以通过把一块标准砖以交叉的位置放在另一块标准砖上得到。关于任何一个新创建的顶点的局部二维曲面不是来自任何平面的任何函数的图形，无论其方向如何。这是一个温和的例子，通常三维情况会更糟。由于缺乏这种数学工具，即直角坐标下的图形，导致难以在域内和边界处获得所需的估计。高维多面体结构存在于线性规划、通信网络和经济系统建模中。PI还不知道类似于刚才描述的椭圆边值问题是否会在任何这些设置中出现，但是边值理论的激励思想，从更小的量中推断出更多的信息，肯定是存在的。

The PI proposes to solve boundary value problems for linear elliptic differential equations and systems in polyhedral domains of n-dimensional Euclidean space. Data is prescribed in Lebesgue spaces and Sobolev spaces defined on the boundary and taken on in the sense of pointwise nontangential limits. The order of the differential equation is not restricted. Polyhedral domains with manifold boundary provide examples of domains for which local continuous transverse vector fields do not exist; i.e. the boundary cannot be locally realized as the graph of a function in rectangular coordinates. This is true in as low a dimension as n=3. This fact poses difficulties in obtaining a priori energy estimates on the boundary (Rellich identities) especially for systems and higher order equations. When such estimates can be obtained, existence can be shown for a closed subspace of data. That the subspace is the entire Banach space of data uses a method of continuity that seems to require a combinatorial structure of the polyhedral boundary. A theorem of E. E. Moise provides this structure in 4 dimensions, but examples in higher dimensions show its lack in general. The PI proposes to find methods of computation to overcome this problem, among others.Two standard examples of linear elliptic boundary value problems are (1) deriving the steady-state temperature distribution inside a solid body from knowledge of the distribution on the boundary (the data), and (2) deriving the systems of stresses, strains and displacements inside an elastic body from knowledge of the boundary's displacement or prescribed stresses at the boundary. In 3-dimensional space there are naturally-occurring physical objects which take polyhedral form (crystal structures, for example). In fact, it is reasonable to think that this occurs more frequently than taking shapes that are either infinitely smooth or as infinitely rough as a boundary surface described by a Lipschitz function. That polyhedral domains can have boundaries not describable as graphs of functions is seen by placing one standard brick upon another in a crossed position. The 2-dimensional surface locally about any one of the newly created vertices is not the graph of any function from any plane no matter how oriented. This is a mild example and in general the 3-dimensional situation can be much worse. The absence of this mathematical tool, the graph in rectangular coordinates, causes difficulties in obtaining the required estimates inside the domain and up to the boundary. Abstract polyhedral structures in higher dimensions are present in linear programming, in modeling communications networks and economic systems. The PI does not yet know if elliptic boundary value problems similar to the ones just described arise in any of these settings, but the motivating idea of boundary value theory, to deduce more information from a smaller amount, is certainly present.