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

本文继续对椭圆系统齐次方程和高阶方程的边值问题进行了数学研究。这项工作强调了边界缺乏平滑假设的领域，这种假设在此类研究中经常被假设。正是这种缺乏平滑性（例如，允许有角和边），为将结果直接应用于具体的物理问题提供了途径，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，给出的边界数据属于各种函数类，如Lebesgue空间、Hardy空间、BMO和Sobolev空间。用边界积分方程给出了解。然而，由于边界缺乏平滑性，所得到的积分方程必须在不借助经典Fredholm理论的情况下求解。某些技术的发展使得双placian等高阶算子和某些系统的Agmon-Miranda型最大原理结果得到了显著改善。具体地说，我们想用沿边界的梯度来估计解的梯度积分。估计是独立于边界函数，应该只依赖于领域的边界的形状。这就导致了奇异积分，需要新的技术来处理。对于双placian，非常一般的结果不能推广到更高的维度，这将是这个项目的主要目标是找到正确的估计，为高维情况下的单个方程以及系统。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 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.