Dimension of p Harmonic Measure and Related Topics

本文研究了在域边界上连续消失的p拉普拉斯方程（常称为p调和函数）的正解所对应的测度维数及其相关问题。如果p是2，p拉普拉斯方程化为拉普拉斯方程，拉普拉斯方程的格林函数，极点在定义域的一个固定点，给出谐波测量。在整个二十世纪，这一措施在势理论和应用领域有许多应用。20世纪80年代中期，数学家开始研究谐波测度的豪斯多夫维数。在一篇重要的论文中，Makarov证明了单连通平面域上的调和测度总是维数为1。Jones和Wolf证明了该测度的维数在任何平面域中总是小于或等于1。当p介于1和无穷之间时，本文给出了p谐波测度的端点型类似的Makarov工作。在这个建议中，他讨论了在任意平面域和高维欧几里德空间的某些域上研究p谐波测度的可能技术。拉普拉斯方程在整个19世纪和20世纪被广泛用于解释物理过程。它的非线性亲戚，p拉普拉斯函数，直到最近才在数学建模（冰川形成，图像处理）中得到应用，也许是因为这种PDE很难使用。PI与合作者的工作使p拉普拉斯在以前为拉普拉斯保留的领域具有很强的可视性。PI和合作者开发了一个p谐波工具箱，使他们能够在以前认为只能用拉普拉斯方程解决的问题上取得重大进展。PI相信在他的建议中讨论的技术、结果和问题将在推广p拉普拉斯算子方面发挥重要作用，并且在数学建模方面也有应用。这些问题技术涉及到调和分析、复函数理论和偏微分方程的良好混合，因此应该对数学和应用分析领域的研究人员和研究生有吸引力。

This proposal is concerned with the dimension of a measure associated with a positive solution to the p Laplace equation, often called a p harmonic function, which vanishes continuously on the boundary of a domain and related problems. If p is two the p Laplace equation reduces to Laplace's equation and the Green's function for Laplace's equation, with pole at a fixed point in the domain, gives harmonic measure. This measure has had numerous applications in Potential Theory and applied areas throughout the twentieth century. In the mid 1980's mathematicians began to study the Hausdorff dimension of harmonic measure. In an important paper, Makarov proved that harmonic measure in simply connected planar domains always has dimension one. Jones and Wolf showed that the dimension of this measure is always less than or equal to one in any planar domain. The proposer has obtained endpoint type analogues of Makarov's work for p harmonic measures, when p is between one and infinity. In this proposal he discusses possible techniques for studying p harmonic measures in arbitrary planar domains and also in certain domains in higher dimensional Euclidean space. Laplace's equation was used widely throughout the nineteenth and twentieth centuries to explain physical processes. Its nonlinear cousin, the p Laplacian, has only recently found applications in mathematical modeling (glacier formation, image processing) perhaps because this PDE is difficult to work with. The PI's work with coauthors gives the p Laplacian strong visibility in an area previously reserved for the Laplacian.The PI and coauthors have developed a p harmonic toolbox which enabled them to make significant progress on problems previously considered solvable only for Laplace's equation. The PI believes that the techniques, results, and problems discussed in his proposal will play an important role in popularizing the p Laplacian and also have applications in mathematical modeling. These problems - techniques involve a nice mixture of harmonic analysis, complex function theory, and partial differential equations, so should be attractive to researchers and graduate students in a wide area of mathematical and applied analysis.