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2017 Structure of sets which are well approximated by zero sets of harmonic polynomials
Matthew Badger, Max Engelstein, Tatiana Toro
Anal. PDE 10(6): 1455-1495 (2017). DOI: 10.2140/apde.2017.10.1455

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 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.

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Matthew Badger. Max Engelstein. Tatiana Toro. "Structure of sets which are well approximated by zero sets of harmonic polynomials." Anal. PDE 10 (6) 1455 - 1495, 2017. https://doi.org/10.2140/apde.2017.10.1455

Information

Received: 1 February 2017; Accepted: 24 April 2017; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 1369.33017
MathSciNet: MR3678494
Digital Object Identifier: 10.2140/apde.2017.10.1455

Subjects:
Primary: 33C55 , 49J52
Secondary: 28A75 , 31A15 , 35R35

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

Rights: Copyright © 2017 Mathematical Sciences Publishers

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Vol.10 • No. 6 • 2017
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