Dimension of p Harmonic Measure and Related Topics

p 谐波测量的维数及相关主题

• 批准号: 1265996
• 负责人: John Lewis
• 金额: $ 13.6万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1265996&HistoricalAwards=false
• 关键词: Dimension; Harmonic; Measure; Related; Topics

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.

这一建议涉及与p-Laplace方程(通常称为p-调和函数)的正解相关的度量的维度，该函数在区域边界上连续消失以及相关问题。若p为2，则p拉普拉斯方程化为拉普拉斯方程，极点在区域内不动点的拉普拉斯方程的格林函数给出调和测度。在整个二十世纪，这一衡量标准在势能理论和应用领域有许多应用。20世纪80年代中期，数学家S开始研究调和测度的Hausdorff维。在一篇重要的文章中，Makarov证明了单连通平面区域上的调和测度总有一维。Jones和Wolf证明了这一度量在任何平面区域上的维度总是小于或等于1。当p介于1和无穷大之间时，作者得到了Makarov关于p个调和测度的端点型类似结果。在这一建议中，他讨论了在任意平面区域以及高维欧氏空间中的某些区域中研究p调和测度的可能方法。拉普拉斯方程在整个十九世纪和二十世纪被广泛用于解释物理过程。它的非线性表亲p-拉普拉斯方程直到最近才在数学建模(冰川形成、图像处理)中得到应用，可能是因为这种偏微分方程组很难处理。PI和合著者共同开发了一个p调和工具箱，使他们能够在以前认为只有拉普拉斯方程可解的问题上取得重大进展。PI相信，他的建议中讨论的技术、结果和问题将在推广p-Laplace方法方面发挥重要作用，并在数学建模中也有应用。这些问题-技术涉及调和分析、复函数理论和偏微分方程的很好的混合，因此应该对数学和应用分析的广泛领域的研究人员和研究生有吸引力。