Flat points in zero sets of harmonic polynomials and harmonic measure from two sides

We obtain quantitative estimates of local flatness of zero sets of harmonic polynomials. There are two alternatives: at every point either the zero set stays uniformly far away from a hyperplane in the Hausdorff distance at all scales or the zero set becomes locally flat on small scales with arbitrarily small constant. An application is given to a free boundary problem for harmonic measure from two sides, where blow-ups of the boundary are zero sets of harmonic polynomials.Comment: 27 pages, 6 figures (in version 2: updated captions and figures, fixed typos and corrected constant in Lemma 4.1

A Two-Phase Free Boundary Problem for Harmonic Measure

We study a 2-phase free boundary problem for harmonic measure first considered by Kenig and Toro and prove a sharp H\"older regularity result. The central difficulty is that there is no a priori non-degeneracy in the free boundary condition. Thus we must establish non-degeneracy by means of monotonicity formulae.Comment: 45 pages. This version has minor revisions as suggested by the refere

Singular points of H\"older asymptotically optimally doubling measures

We consider the question of how the doubling characteristic of a measure determines the regularity of its support. The question was considered by David, Kenig, and Toro for codimension-1 under a crucial assumption of flatness, and later by Preiss, Tolsa, and Toro in higher codimension. However, their studies leave open the geometry of the support of such measures in a neighborhood about a non-flat point of the support. We here answer the question (in an almost classical sense) for codimension-1 H\"older doubling measures in \RR^4.Comment: 43 pages, 7 figure

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

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.Comment: 40 pages, 2 figures (v2: streamlined several proofs, added statement of Lojasiewicz inequality for harmonic polynomials [Theorem 3.1]

Local set approximation: Mattila-Vuorinen type sets, Reifenberg type sets, and tangent sets

We investigate the interplay between the local and asymptotic geometry of a set $A \subseteq \mathbb{R}^n$ and the geometry of model sets $\mathcal{S} \subset \mathcal{P}(\mathbb{R}^n)$, which approximate $A$ locally uniformly on small scales. The framework for local set approximation developed in this paper unifies and extends ideas of Jones, Mattila and Vuorinen, Reifenberg, and Preiss. We indicate several applications of this framework to variational problems that arise in geometric measure theory and partial differential equations. For instance, we show that the singular part of the support of an $(n-1)$-dimensional asymptotically optimally doubling measure in $\mathbb{R}^n$ ($n\geq 4$) has upper Minkowski dimension at most $n-4$.Comment: 52 pages, 5 figure