## Grants and Contracts Details

### Description

Project Summary
This proposal discusses two kinds of problems. The first kind involves existence and uniqueness
in some overdetermined free boundary problems of p Laplacian type under minimal boundary
assumptions. The proposer, coauthors, and his PhD student, have obtained existence and
uniqueness in certain of these problems. The proposer would like to investigate the sharpness
of these results in higher dimensions, as well as lower dimensional analogues, and also consider
parabolic generalizations. Problems of the second kind are concerned with the dimension of a
measure associated with a positive p harmonic function, vanishing on the boundary of a certain
domain. The proposer and his PhD student have obtained analogues of well known theorems for
harmonic measure in a quasicirde. The PI would like to investigate extensions of these theorems
to simply connected domains and planar domains in general. Related questions involve a rate
theorem for p harmonic functions and estimates of the dimension as a function of p.
Intellectual Merit: The problems in the proposal are simply stated and the PI believes, quite
natural and fundamental questions to ask. However there solutions are often rather complicated
and appear related to some fundamental questions in harmonic analysis and function theory. For
example, the Riesz transforms problem, characterizations of uniform rectifiability, the dimension
and absolute continuity of elliptic measures, are all questions which arise naturally in our existence,
uniqueness, dimension, problems. So the proposer's research could also lead to solutions
or new insights into these related questions.
Broad Impact: The PI notes that the p Laplacian is often used in mathematical modeling and
free boundary - inverse type problems occur naturally in nature. Also existence and uniqueness
are important questions in applications and form one of the core areas in partial differential
equations. In addition to possible applications, the problems in this proposal are a good mix of
harmonic analysis, function theory, and partial differential equations, so should be attractive to
researchers and graduate students who are interested in classical analysis. The PI is committed to
working with graduate students, through one on one lectures, and joint written collaborations.
Through his efforts he hopes to produce classical analysts that are capable of doing quality
research at a high level.
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Status | Finished |
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Effective start/end date | 7/1/06 → 6/30/10 |

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