Dimension of p Harmonic Measure and Related Topics

这项建议涉及到 与p拉普拉斯方程的正解相关的测度的维数，通常称为p调和函数，它在区域和相关问题的边界上连续为零。当p为2时，p的拉普拉斯方程化为拉普拉斯方程，而极点在区域内不动点的拉普拉斯方程的绿色函数给出调和测度. 这一措施在许多国家得到了应用， 潜能理论及其应用领域贯穿整个世纪。 80年代中期 数学家们开始研究调和测度的Hausdorff维数。 在一篇重要的论文中， 马卡罗夫 证明了平面单连通区域上的调和测度总是一维的。 Jones和Wolf证明了这个测度的维数在任何平面域中总是小于或等于1。 当p在1和无穷大之间时，提出者得到了Makarov关于p调和测度的工作的端点型类似物。 在这个建议中，他讨论了可能的技术研究p谐波措施， 任意平面域，也在高维欧氏空间中的某些域中。 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。它的非线性表亲，p拉普拉斯算子，最近才发现在数学建模（冰川形成，图像处理）中的应用，也许是因为这个偏微分方程很难处理。 PI与合著者的工作使p Laplacian在以前为Laplacian保留的区域中具有很强的可见性。PI和合著者开发了一个p谐波工具箱，使他们能够在以前认为只能为拉普拉斯方程解决的问题上取得重大进展。 PI认为，他的提案中讨论的技术、结果和问题将 对推广p-Laplacian有重要作用，在数学建模中也有应用。这些问题-技术涉及调和分析，复变函数理论和偏微分方程的很好的混合，所以应该是有吸引力的研究人员和研究生在广泛的数学和应用分析领域。

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.