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

Uniqueness of the blow-up at isolated singularities for the Alt-Caffarelli functional

In this paper we prove uniqueness of blow-ups and $C^{1,\log}$-regularity for the free-boundary of minimizers of the Alt-Caffarelli functional at points where one blow-up has an isolated singularity. We do this by establishing a (log-)epiperimetric inequality for the Weiss energy for traces close to that of a cone with isolated singularity, whose free-boundary is graphical and smooth over that of the cone in the sphere. With additional assumptions on the cone, we can prove a classical epiperimetric inequality which can be applied to deduce a $C^{1,\alpha}$ regularity result. We also show that these additional assumptions are satisfied by the De Silva-Jerison-type cones, which are the only known examples of minimizing cones with isolated singularity. Our approach draws a connection between epiperimetric inequalities and the \L ojasiewicz inequality, and, to our knowledge, provides the first regularity result at singular points in the one-phase Bernoulli problem.Comment: 37 pages. To appear in Duke Math Journa

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]

(Log-)epiperimetric inequality and regularity over smooth cones for almost Area-Minimizing currents

We prove a new logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularity by flowing in the radial direction any given trace along appropriately chosen directions. In contrast to previous epiperimetric inequalities for minimal surfaces (e.g. those of Reifenberg, Taylor and White), we need no a priori assumptions on the structure of the cone (e.g. integrability). Moreover, if the cone is integrable (not only through rotations), we recover the classical epiperimetric inequality. As a consequence we deduce a new $\varepsilon$-regularity result for almost area-minimizing currents at singular points, where at least one blow-up is a multiplicity-one cone with isolated singularity. This result is similar to the one for stationary varifolds of Leon Simon, but independent from it since almost minimizers do not satisfy any equation