Multidirectional Boundry Value Problems

PI提出了在n维欧氏空间的多面体区域上求解线性椭圆型微分方程组的边值问题。数据是在定义在边界上的勒贝格空间和索博列夫空间中规定的，并且在逐点非切向极限的意义下被采用。微分方程式的阶数不受限制。具有流形边界的多面体区域提供了不存在局部连续横向向量场的区域的例子；即边界不能局部地实现为直角坐标下的函数的图形。在n=3的低维情况下也是如此。这一事实给在边界上获得先验能量估计(Rellich恒等式)带来了困难，特别是对于系统和高阶方程。当可以获得这样的估计时，可以证明数据的闭子空间的存在性。子空间是数据的整个Banach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4个维度上提供了这种结构，但更高维度的例子表明它总体上是不存在的。PI建议寻找计算方法来克服这个问题。两个标准的线性椭圆边值问题的例子是(1)根据边界上的分布(数据)导出固体内部的稳态温度分布，以及(2)根据边界上的位移或规定的应力导出弹性体内的应力、应变和位移系统。在3维空间中，自然存在的物理对象呈多面体形式(例如晶体结构)。事实上，我们有理由认为，这种情况比采用无限光滑或无限粗糙的形状更常见，这些形状要么是无限光滑的，要么是由Lipschitz函数描述的边界表面。通过将一块标准砖放在另一块砖上的交叉位置，可以看到多面体区域可以具有不能像函数图那样描述的边界。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，一般来说，3维的情况可能会更糟。缺少这种数学工具，即直角坐标下的图形，就很难在域内和边界内获得所需的估计值。摘要高维多面体结构存在于线性规划、通信网络和经济系统的建模中。PI还不知道类似于刚才描述的椭圆边值问题是否出现在这些设置中的任何一个，但边值理论的激励思想，从较小的数量推导出更多的信息，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.