Flat points in zero sets of harmonic polynomials and harmonic measure from two sides

We obtain quantitative estimates of local flatness of zero sets of harmonic polynomials. There are two alternatives: at every point either the zero set stays uniformly far away from a hyperplane in the Hausdorff distance at all scales or the zero set becomes locally flat on small scales with arbitra...

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Bibliographic Details
Published in:Journal of the London Mathematical Society 2013-02, Vol.87 (1), p.111-137
Main Author: Badger, Matthew
Format: Article
Language:English
Subjects:
Boundaries
Constants
Flats
Free boundaries
Harmonics
Mathematical analysis
Polynomials
Small scale
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Summary:We obtain quantitative estimates of local flatness of zero sets of harmonic polynomials. There are two alternatives: at every point either the zero set stays uniformly far away from a hyperplane in the Hausdorff distance at all scales or the zero set becomes locally flat on small scales with arbitrarily small constant. An application is given to a free boundary problem for harmonic measure from two sides, where blow‐ups of the boundary are zero sets of harmonic polynomials.
ISSN:0024-6107
1469-7750
DOI:10.1112/jlms/jds041