Problems of Existence, Uniqueness, and Dimension in Harmonic Analysis, Function Theory, and Partial Differential Equations

调和分析、函数论和偏微分方程中的存在性、唯一性和维数问题

• 批准号: 0552281
• 负责人: John Lewis
• 金额: $ 13.5万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0552281&HistoricalAwards=false
• 关键词: Problems; Existence; Uniqueness; Dimension; Harmonic

The proposed research will investigate existence and uniqueness of solutions of some overdetermined free boundary problems of p-Laplacian type under minimal smoothness and boundary assumptions. Similar questions for solutions of parabolic free boundary problems will also be considered. The investigator will also study the dimension of a measure associated with a positive p-harmonic function vanishing on the boundary of a certain domain. Related questions include proving a boundary Harnack inequality for certain p-harmonic functions. The problems on dimension are direct analogues for p-harmonic functions of work of Bishop, Carleson, Jones, Makarov, and Wolff for harmonic functions. These questions about free boundary problems are part of a program initiated by Andrew Vogel and the proposer with the intent of obtaining symmetry or uniqueness theorems under minimal overdetermined boundary assumptions. The expected theorems will generalize results of Serrin in the smooth case and work of Alt and Caffarelli on free boundary problems arising from certain minima in the calculus of variations. These problems also appear related to some fundamental questions in harmonic analysis such as the Riesz transforms problem, characterizations of uniform rectifiability, and absolute continuity of elliptic measure

本文主要研究一类p-Laplacian型超定自由边界问题在极小光滑性和边界假设下解的存在唯一性。抛物自由边界问题的解的类似问题也将被考虑。 研究者还将研究与在某个域的边界上消失的正p-调和函数相关的测度的维数。相关的问题包括证明某些p-调和函数的边界Harnack不等式。 维数问题是直接类似的p-调和函数的工作主教，Carleson，琼斯，马卡罗夫，和沃尔夫调和函数。 这些关于自由边界问题的问题是安德鲁·沃格尔和提出者发起的一个计划的一部分，目的是在最小超定边界假设下获得对称性或唯一性定理。预期的定理将推广结果Serrin在顺利的情况下，工作的Alt和Caffarelli自由边界问题所产生的某些极小的变分法。这些问题还涉及到调和分析中的一些基本问题，如Riesz变换问题、一致可求长性的刻画、椭圆测度的绝对连续性等