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

对于大多数p值，p-Laplacian是一个非线性发散形式的退化椭圆型偏微分方程. 解决这个 PDE （称为p调和函数）是相对好的，因为它们在旋转，平移，膨胀下是不变的，并且在乘以常数时也保持解。 尽管如此，这种偏微分方程的非线性使得即使是基本的问题也难以回答。最近，主要研究者和合著者Nystrom证明了一个边界Harnack不等式（包括保持器连续性），该不等式是关于两个正p调和函数之比在Lipschitz域的一部分上消失的. 本项目建议书涉及这一应用 边界Harnack不等式应用于p-Martin边界、p-调和测度的维数以及某些两相自由边界问题. 大多数物理模型涉及线性偏微分方程（在其主要部分）。 拉普拉斯方程是最著名的线性偏微分方程之一，常用于数学模型和描述物理现象。本文将拉普拉斯方程的一些经典结果推广到非线性p拉普拉斯方程。提议者和合著者最近开发的技术使这种扩展成为可能。 希望我们的工作最终能使p拉普拉斯方程在数学建模和一般科学中得到更广泛的应用。

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.