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

本文继续对椭圆组齐次方程边值问题和高阶方程边值问题进行数学研究。这项工作强调了边界缺乏光滑性假设的领域，这些假设通常在此类研究中假设。正是这种不光滑性(例如，允许拐角和边缘)为将结果直接应用于具体的物理问题提供了一条途径，在这些问题中，相对粗糙的边界是规则而不是例外。这些方程是多维的，给出了属于各种函数类的边界数据，如勒贝格空间、哈代空间、BMO空间和索伯列夫空间。给出了用边界积分方程组表示的解。然而，由于边界缺乏光滑性，由此产生的积分方程组必须求解，而不能求助于经典的Fredholm理论。对于高阶算符，如双盘系统和某些系统，已经发展了某些技术，这些技术已经显著地改进了Agmon-Miranda类型的最大原理结果。具体地说，我们想用沿边界的梯度来估计解的梯度的积分。估计值应独立于边界函数，并且应仅取决于区域边界的形状。这导致了需要新技术的奇异积分。对于双普拉斯方程，已经在二维和三维得到了非常普遍的结果。这些方法不能扩展到更高的维，这将是这个项目的主要目标是为高维的情况找到正确的估计，无论是对单方程还是对系统。这类方程的典型来源是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 assumptions 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 function and should only depend on the shape of the domain's boundary. This leads to singular integrals which require new techniques. For the bilaplacian, very general results have been obtained in two and three dimensions. The methods 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 hydrostatics and electrostatics.