Mathematical Sciences: Elliptic Boundary Value Problems and Maximum Principles on Nonsmooth Domains

点击翻译按钮获取中文摘要

This work continues mathematical research on boundary value problems for homogeneous equations of elliptic systems and higher order equations. The work emphasizes domains whose boundaries lack the smoothness assumption often assumed in such studies. It is this lack of smoothness (corners and edges are allowed, for example), which provides an avenue for direct application of the results to concrete physical problems where relatively rough boundaries are the rule rather than the exception. The equations are multi-dimensional, given with boundary data belonging to various function classes such as the Lebesgue spaces, Hardy spaces, BMO and Sobolev spaces. Solutions are given in terms of boundary integral equations. However, because the boundaries lack smoothness, the resulting integral equations must be solved without recourse to the classical Fredholm theory. Certain techniques have been developed which have led to significantly improved maximum principle results of Agmon-Miranda type for higher order operators such as the bilaplacian and certain systems. Specifically, one wants to estimate integrals of gradients of solutions in terms of the gradient along the boundary. The estimates are to be independent of the boundary functions and should only depend on the shape of the domain's boundary. This leads to singular integrals which require new techniques to handle. For the bilaplacian, very general results cannot be extended to higher dimensions, and it will be the primary goal of this project to find the correct estimates for the higher dimensional cases both for single equations as well as for systems. Typical sources for such equations are from hydrostatics and electrostatics.