## Analysis & PDE

• Anal. PDE
• Volume 10, Number 6 (2017), 1455-1495.

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

#### 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 $ℝn$ 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.

#### Article information

Source
Anal. PDE, Volume 10, Number 6 (2017), 1455-1495.

Dates
Accepted: 24 April 2017
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.apde/1510843529

Digital Object Identifier
doi:10.2140/apde.2017.10.1455

Mathematical Reviews number (MathSciNet)
MR3678494

Zentralblatt MATH identifier
1369.33017

#### Citation

Badger, Matthew; Engelstein, Max; Toro, Tatiana. Structure of sets which are well approximated by zero sets of harmonic polynomials. Anal. PDE 10 (2017), no. 6, 1455--1495. doi:10.2140/apde.2017.10.1455. https://projecteuclid.org/euclid.apde/1510843529

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