Global Boundary Value Problems With Minimal Smoothness Assumptions

拟议的研究涉及一般， 椭圆型和抛物型的变系数方程组。 将特别注意该理论的几个关键特征。 第一，着重强调具有全球性质的问题。 特别是，底层流形的拓扑预计将发挥作用， 一个重要的角色。此外，全局积分表示公式（以 的多层类型运营商）的解决方案。 第二，对所涉及的分析结构和几何结构进行最小平滑度假设。在这方面，一个象征性的 微积分不再是现成的，因此， 手头的问题已经大大改变了。 总的目标是通过调和分析和偏微分方程的现代工具进行系统的研究，这样的问题。 这样的问题不仅是纯粹的学术兴趣。除了单纯的 理解理论的自然局限性，这项研究也是 由现实生活中的问题（如涉及边缘域的问题， 拐角或裂缝，不连续系数和/或边界数据， 非均匀和/或各向异性介质），其中非光滑问题比光滑问题多得多。 实际上，任何实际的应用，例如计算 从飞机的身体散射波，将不得不面对 某种粗糙度，例如具有非光滑边界的域。形成了一个理论， 为这些问题设计有效的数值算法将是一个 提出研究的主要目标。

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.