### 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 ≥ 2 and 1 ≤ k ≤ 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 ≥ 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.

Original language | English (US) |
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Pages (from-to) | 1455-1495 |

Number of pages | 41 |

Journal | Analysis and PDE |

Volume | 10 |

Issue number | 6 |

DOIs | |

State | Published - Jan 1 2017 |

Externally published | Yes |

### Keywords

- Harmonic measure
- Harmonic polynomials
- Hausdorff dimension
- Lojasiewicz-type inequalities
- Minkowski dimension
- NTA domains
- Reifenberg-type sets
- Singular set
- Two-phase free boundary problems

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## Cite this

*Analysis and PDE*,

*10*(6), 1455-1495. https://doi.org/10.2140/apde.2017.10.1455