Analysis & PDE
- Anal. PDE
- Volume 10, Number 6 (2017), 1455-1495.
Structure of sets which are well approximated by zero sets of harmonic polynomials
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- points” sit inside zero sets of harmonic polynomials in of degree (for all and ) 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- points () 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 . An application is given to the two-phase free boundary regularity problem for harmonic measure below the continuous threshold introduced by Kenig and Toro.
Anal. PDE, Volume 10, Number 6 (2017), 1455-1495.
Received: 1 February 2017
Accepted: 24 April 2017
First available in Project Euclid: 16 November 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 33C55: Spherical harmonics 49J52: Nonsmooth analysis [See also 46G05, 58C50, 90C56]
Secondary: 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15] 31A15: Potentials and capacity, harmonic measure, extremal length [See also 30C85] 35R35: Free boundary problems
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