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

摘要

The zero sets of harmonic polynomials play a crucial role in the study of thefree boundary regularity problem for harmonic measure. In order to understandthe fine structure of these free boundaries a detailed study of the singularpoints 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 admitarbitrarily good local approximations by zero sets of harmonic polynomials. Weobtain a general structure theorem for the latter type of sets, including sharpHausdorff and Minkowski dimension estimates on the singular set of "degree $k$points" ($k\geq 2$) without proving uniqueness of blowups or aid of PDE methodssuch as monotonicity formulas. In addition, we show that in the presence of acertain topological separation condition, the sharp dimension estimates improveand depend on the parity of $k$. An application is given to the two-phase freeboundary regularity problem for harmonic measure below the continuous thresholdintroduced by Kenig and Toro.