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

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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