Applications of Boundary Harnack Inequalities for p Harmonic Functions to Problems in Harmonic Analysis, PDE, and Function Theory

对于p的大多数值，p的拉普拉斯算子是退化椭圆型偏微分方程解的非线性发散形式。这种偏微分方程解(称为p调和函数)相对较好，因为它们在旋转、平移、伸缩时是不变的，在乘以常量时也是不变的。尽管如此，这种偏微分方程的非线性使得甚至连基本的问题都很难回答。最近，主要研究者和合著者Nystrom证明了关于在Lipschitz区域的一部分上消失的两个正p调和函数之比的边界Harnack不等式(包括Holder连续性)。本项目的目的是将这一边界Harnack不等式应用于p-Martin边界、p-调和测度的维度问题，以及某些两相自由边界问题。大多数物理模型都包含线性偏微分方程组(在它们的主体部分)。拉普拉斯方程是最著名的线性偏微分方程组之一，常用于数学模型和描述物理现象。这一建议涉及将拉普拉斯方程的一些经典结果推广到它的近亲--非线性p-拉普拉斯方程。发起人和合著者开发的最新技术使这一扩展现在成为可能。我们希望我们的工作最终将导致在数学建模和一般的科学中更多地使用p-Laplace方程。

The p Laplacian for most values of p is a nonlinear divergence form degenerate elliptic PDE. Solutions to this PDE (called p harmonic functions) are relatively nice in that they are invariant under rotations, translations, dilations, and also remain solutions under multiplication by constants. Still the nonlinearity of this PDE makes even basic questions difficult to answer. Recently the principal investigator and coauthor Nystrom have proved a boundary Harnack inequality (including Holder continuity) for the ratio of two positive p harmonic functions vanishing on a portion of a Lipschitz domain. This project proposal is concerned with applications of this boundary Harnack inequality to problems concerning the p Martin boundary, the dimension of p harmonic measure, and to certain two phase free boundary problems. Most physical models involve linear PDE (in their principal part). Laplace`s equation is one of the best known linear PDE and is often used in mathematical models, as well as to describe physical phenomena. This proposal is concerned with extending some classical results for Laplace's equation to its cousin the nonlinear p Laplace equation. Recent technology developed by the proposer and coauthors make this extension now possible. It is hoped that our work will eventually lead to greater use of the p Laplace equation in mathematical modeling and in general in the sciences.