Flat points in zero sets of harmonic polynomials and harmonic measure from two sides
Classical Analysis and ODEs
2013-05-22 v2 Analysis of PDEs
Abstract
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.
Cite
@article{arxiv.1109.1427,
title = {Flat points in zero sets of harmonic polynomials and harmonic measure from two sides},
author = {Matthew Badger},
journal= {arXiv preprint arXiv:1109.1427},
year = {2013}
}
Comments
27 pages, 6 figures (in version 2: updated captions and figures, fixed typos and corrected constant in Lemma 4.1)