Slowly vanishing mean oscillations: non-uniqueness of blow-ups in a two-phase free boundary problem

TL;DR

This paper presents examples of domains where the boundary's blow-up limits are non-unique despite the logarithm of the Radon-Nikodym derivative being continuous, highlighting subtle irregularities in free boundary problems.

Contribution

It constructs domains with continuous log h where boundary blow-ups are non-unique, revealing new complexities in the regularity theory of free boundaries.

Findings

Existence of domains with non-unique blow-ups despite continuous log h

Slow oscillation or rotation of blow-up limits causes non-uniqueness

Highlights subtle irregularities in free boundary regularity theory

Abstract

In Kenig and Toro's two-phase free boundary problem, one studies how the regularity of the Radon-Nikodym derivative $h= d\omega^-/d\omega^+$ of harmonic measures on complementary NTA domains controls the geometry of their common boundary. It is now known that $\log h \in C^{0,\alpha}(\partial \Omega)$ implies that pointwise the boundary has a unique blow-up, which is the zero set of a homogeneous harmonic polynomial. In this note, we give examples of domains with $\log h \in C(\partial \Omega)$ whose boundaries have points with non-unique blow-ups. Philosophically the examples arise from oscillating or rotating a blow-up limit by an infinite amount, but very slowly.