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Badger, Matthew; Engelstein, Max; Toro, Tatiana

Structure of sets which are well approximated by zero sets of harmonic polynomials. (English) Zbl 1369.33017

Anal. PDE 10, No. 6, 1455-1495 (2017).
Summary: 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 \(\mathbb{R}^n\) 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.

Cited in 7 Documents

MSC:

33C55 Spherical harmonics
28A75 Length, area, volume, other geometric measure theory
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions

Keywords:

Reifenberg-type sets; harmonic polynomials; Łojasiewicz-type inequalities; singular set; Hausdorff dimension; Minkowski dimension; two-phase free boundary problems; harmonic measure; NTA domains
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\textit{M. Badger} et al., Anal. PDE 10, No. 6, 1455--1495 (2017; Zbl 1369.33017)
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