Vol. 10, No. 6, 2017

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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
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.

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