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Structure of sets
which are well approximated by zero sets of harmonic
polynomials
Matthew Badger, Max Engelstein and Tatiana Toro |
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Vol. 10 (2017), No. 6, 1455–1495
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Abstract |
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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- 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. |
Keywords
Reifenberg-type sets, harmonic polynomials,
\Lojasiewicz-type inequalities, singular set, Hausdorff
dimension , Minkowski dimension, two-phase free boundary
problems, harmonic measure, NTA domains
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Mathematical Subject Classification 2010
Primary: 33C55, 49J52
Secondary: 28A75, 31A15, 35R35
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Milestones
Received: 1 February 2017
Accepted: 24 April 2017
Published: 14 July 2017
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Authors |
| Matthew Badger | |
| Department of Mathematics University of Connecticut Storrs, CT 06269-1009 United States |
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| Max Engelstein | |
| Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139-4307 United States |
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| Tatiana Toro | |
| Department of Mathematics University of Washington Box 354350 Seattle, WA 98195-4350 United States |
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