Regularity of the singular set in a two-phase problem for harmonic measure with Hölder data

Abstract

In non-variational two-phase free boundary problems for harmonic measure, we examine how the relationship between the interior and exterior harmonic measures of a domain $\Omega \subset {\mathbb{R}}^n$ influences the geometry of its boundary. This type of free boundary problem was initially studied by Kenig and Toro in 2006, and was further examined in a series of separate and joint investigations by several authors. The focus of the present paper is on the singular set in the free boundary, where the boundary looks infinitesimally like zero sets of homogeneous harmonic polynomials of degree at least 2. We prove that if the Radon–Nikodym derivative of the exterior harmonic measure with respect to the interior harmonic measure has a Hölder continuous logarithm, then the free boundary admits unique geometric blowups at every singular point and the singular set can be covered by countably many $C^{1,\beta }$ submanifolds of dimension at most $n-3$. This result is partly obtained by adapting tools such as Garofalo and Petrosyan’s Weiss type monotonicity formula and an epiperimetric inequality for harmonic functions from the variational to the non-variational setting.