Multidirectional Boundry Value Problems

PI提出在n维欧氏空间的多面体域上求解线性椭圆型微分方程和系统的边值问题。 数据是规定在勒贝格空间和Sobolev空间定义的边界上，并采取了意义上的逐点非切向限制。 微分方程的阶数不受限制。 具有流形边界的多面体域提供了局部连续横向向量场不存在的域的例子;即边界不能局部地实现为直角坐标中的函数的图。 这在n=3的低维中是正确的。 这一事实造成的困难，在获得先验的能量估计的边界（Rellich身份），特别是系统和高阶方程。 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。 子空间是整个数据的Banach空间，使用的连续性方法似乎需要多面体边界的组合结构。 E. E. Moise在4维中提供了这种结构，但在更高维度中的例子表明它通常缺乏。 PI提出寻找计算方法来克服这个问题，其中，线性椭圆边值问题的两个标准例子是：（1）从边界上的分布知识导出固体内部的稳态温度分布（数据），和（2）推导应力系统，根据边界位移或边界处的规定应力的知识来计算弹性体内部的应变和位移。 在三维空间中，有自然存在的物理对象，它们采取多面体形式（例如晶体结构）。 事实上，我们有理由认为，这种情况发生的频率要比采用无限光滑或无限粗糙的形状（如Lipschitz函数所描述的边界曲面）更高。 多面体域可以有无法用函数图描述的边界，这可以通过将一块标准砖放在另一块标准砖上的交叉位置来看到。 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，无论如何定向。 这是一个温和的例子，一般来说，三维情况可能更糟。 由于没有这种数学工具，即直角坐标图，因此难以在域内和边界上获得所需的估计数。 高维抽象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还不知道在这些设置中是否会出现类似于刚才描述的椭圆边值问题，但是边值理论的激励思想，从更少量中推导出更多的信息，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.