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

Badger, Matthew; Engelstein, Max; Toro, Tatiana

doi:10.2140/apde.2017.10.1455

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

Matthew Badger, Max Engelstein and Tatiana Toro

Vol. 10 (2017), No. 6, 1455–1495

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 $ℝ^{n}$ of degree $d$ (for all $n \geq 2$ and $1 \leq k \leq d$ ) 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 ( $k \geq 2$ ) 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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Keywords

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

Mathematical Subject Classification 2010

Primary: 33C55, 49J52

Secondary: 28A75, 31A15, 35R35

Milestones

Received: 1 February 2017

Accepted: 24 April 2017

Published: 14 July 2017

Authors

Matthew Badger
Department of Mathematics University of Connecticut Storrs, CT 06269-1009 United States

Max Engelstein
Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139-4307 United States

Tatiana Toro
Department of Mathematics University of Washington Box 354350 Seattle, WA 98195-4350 United States

Vol. 10, No. 6, 2017