The proposed research is concerned with the treatment of general, variable coefficient systems of equations of elliptic and parabolic type. Special attention will be paid to several key features of the theory. First, a strong emphasis is placed on problems having a global character. In particular, the topology of the underlying manifold is expected to play a significant role. Also, global integral representation formulas (in terms of multi-layer type operators) for the solution are sought. Second, minimal smoothness assumptions are to be made on the analytical and geometrical structures involved. In this context, a symbolic calculus is no longer readily available and, hence, the nature of the problems at hand is significantly altered. The overall objective is to undertake a systematic study of such problems via the modern tools of harmonic analysis and partial differential equations. Such problems are not only of a purely academic interest. Besides the mere understanding of the natural limits of the theory, this study is also motivated by real-life problems (like those involving domains with edges, corners or cracks, discontinuous coefficients and/or boundary data, non-homogeneous and/or anisotropic media) where non-smooth problems are considerably more abundant than smooth ones. Indeed, any realistic application, such as calculating the scattered wave from the body of an airplane, will have to confront some kind of roughness such as domains with non-smooth boundaries. Formulating the theory which ultimately allows for the design of efficient numerical algorithms for such problems will be one of the main goals of the proposed research.
