Global Boundary Value Problems With Minimal Smoothness Assumptions

所提出的研究涉及一般的变系数椭圆型和抛物型方程组的处理。我们将特别关注这一理论的几个关键特征。首先，对具有全球性的问题给予了高度重视。特别是，预计底层歧管的拓扑结构将发挥重要作用。同时，求出了解的全局积分表示式(用多层型算子表示)。其次，对所涉及的分析和几何结构进行最小光滑度假设。在这种情况下，符号演算不再容易获得，因此，手头问题的性质被显著改变。总体目标是通过调和分析和偏微分方程式等现代工具对这些问题进行系统研究。这样的问题不仅仅是纯粹的学术兴趣。除了对理论的自然极限的单纯理解外，这项研究还受到实际问题的推动(如涉及具有边、角或裂纹的区域、不连续的系数和/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要丰富得多。事实上，任何实际的应用，如计算飞机机身的散射波，都必须面对某种粗糙度，例如具有非光滑边界的区域。形成最终允许为这类问题设计有效的数值算法的理论将是拟议研究的主要目标之一。

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.