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.
