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

大多数p值的拉普拉斯算子是一种非线性散度形式的退化椭圆偏微分方程。这个PDE的解（称为p调和函数）相对来说很好，因为它们在旋转、平动、膨胀下是不变的，并且在乘以常数时仍然是解。然而，这种PDE的非线性甚至使基本问题难以回答。最近，首席研究员和合著者Nystrom证明了两个正p调和函数的比值在Lipschitz域的一部分上消失的边界Harnack不等式（包括Holder连续性）。本课题讨论了该边界哈纳克不等式在p马丁边界、p谐波测度维数问题和某些两相自由边界问题中的应用。大多数物理模型都涉及线性偏微分方程（在其主体部分）。拉普拉斯方程是最著名的线性偏微分方程之一，经常用于数学模型中，也用于描述物理现象。本文讨论了将拉普拉斯方程的一些经典结果推广到它的同类非线性p拉普拉斯方程。由提议者和共同作者开发的最新技术使这种扩展现在成为可能。我们希望我们的工作最终将导致p拉普拉斯方程在数学建模和一般科学中的更多使用。

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.