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

In Kenig and Toro’s two-phase free boundary problem, one studies how the regularity of the Radon-Nikodym derivative $h=d{\omega }^{-}\mathrm{/}d{\omega }^{+}$ of harmonic measures on complementary NTA domains controls the geometry of their common boundary. It is now known that $\log h\in C^{0,\alpha }(\mathrm{\partial }\mathrm{\Omega })$ 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 $\log h\in C(\mathrm{\partial }\mathrm{\Omega })$ 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.