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

研究了一类p- laplace型超定自由边界问题在最小光滑性和边界假设下解的存在唯一性。抛物线自由边界问题解的类似问题也将被考虑。研究者还将研究与消失在某一域边界上的正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调和函数的边界哈纳克不等式。维数问题是Bishop、Carleson、Jones、Makarov和Wolff对调和函数的功的p调和函数的直接类比。这些关于自由边界问题的问题是由Andrew Vogel和提议者发起的一个项目的一部分，目的是在最小的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Serrin在光滑情况下的结果，以及Alt和Caffarelli在变分法中由某些极小值引起的自由边界问题上的工作。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、一致可纠偏性的刻画、椭圆测度的绝对连续性等

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