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

本文继续了边值问题的数学研究 椭圆型方程组的齐次方程问题 阶方程 这项工作强调的领域， 缺乏这类研究中通常假设的平滑性假设。 它 这种缺乏平滑度（允许角和边， 例如），这为直接应用 结果，具体的物理问题，其中相对粗糙 边界是规则而不是例外。 该方程是多维的，有边界 属于各种函数类的数据，例如Lebesgue 空间、哈代空间、BMO空间和Sobolev空间。 解决方案 用边界积分方程表示。 但由于 边界缺乏平滑性，由此产生的积分方程 必须在不求助于经典Fredholm理论的情况下解决。 已经开发出了某些技术， 显著改进了Agmon-Miranda的最大原理结果 高阶算子的类型，如bilaplacian和 某些系统。 具体来说，我们要估计积分 梯度的解决方案，在梯度沿着 边界 估计值与边界无关 函数，应该只取决于域的形状 边界 这导致奇异积分，需要新的 技术来处理。 对于bilaplacian， 不能扩展到更高的维度，它将是 该项目的主要目标是找到正确的估计， 高维情况下，无论是单一的方程，以及 for systems系统. 这些方程的典型来源来自 流体静力学和静电学。

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.