#### Vol. 10, No. 6, 2017

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Structure of sets which are well approximated by zero sets of harmonic polynomials

### Matthew Badger, Max Engelstein and Tatiana Toro

Vol. 10 (2017), No. 6, 1455–1495
##### Abstract

The zero sets of harmonic polynomials play a crucial role in the study of the free boundary regularity problem for harmonic measure. In order to understand the fine structure of these free boundaries, a detailed study of the singular points of these zero sets is required. In this paper we study how “degree-$k$ points” sit inside zero sets of harmonic polynomials in ${ℝ}^{n}$ of degree $d$ (for all $n\ge 2$ and $1\le k\le d$) and inside sets that admit arbitrarily good local approximations by zero sets of harmonic polynomials. We obtain a general structure theorem for the latter type of sets, including sharp Hausdorff and Minkowski dimension estimates on the singular set of degree-$k$ points ($k\ge 2$) without proving uniqueness of blowups or aid of PDE methods such as monotonicity formulas. In addition, we show that in the presence of a certain topological separation condition, the sharp dimension estimates improve and depend on the parity of $k$. An application is given to the two-phase free boundary regularity problem for harmonic measure below the continuous threshold introduced by Kenig and Toro.

##### Keywords
Reifenberg-type sets, harmonic polynomials, \Lojasiewicz-type inequalities, singular set, Hausdorff dimension , Minkowski dimension, two-phase free boundary problems, harmonic measure, NTA domains
##### Mathematical Subject Classification 2010
Primary: 33C55, 49J52
Secondary: 28A75, 31A15, 35R35