Analysis & PDE

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

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

Matthew Badger, Max Engelstein, and Tatiana Toro

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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 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.

Article information

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

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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

Reifenberg-type sets harmonic polynomials \Lojasiewicz-type inequalities singular set Hausdorff dimension Minkowski dimension two-phase free boundary problems harmonic measure NTA domains


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.

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