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

Matthew Badger, Max Engelstein, Tatiana Toro

Research output: Contribution to journalArticle

3 Scopus citations

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 ℝn 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.

Original languageEnglish (US)
Pages (from-to)1455-1495
Number of pages41
JournalAnalysis and PDE
Volume10
Issue number6
DOIs
StatePublished - Jan 1 2017
Externally publishedYes

Keywords

  • Harmonic measure
  • Harmonic polynomials
  • Hausdorff dimension
  • Lojasiewicz-type inequalities
  • Minkowski dimension
  • NTA domains
  • Reifenberg-type sets
  • Singular set
  • Two-phase free boundary problems

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