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

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.

### 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
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### References:

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