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In Kenig and Toro's two-phase free boundary problem, one studies how theregularity of the Radon-Nikodym derivative $h= d\omega^-/d\omega^+$ of harmonicmeasures on complementary NTA domains controls the geometry of their commonboundary. 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 setof a homogeneous harmonic polynomial. In this note, we give examples of domainswith $\log h \in C(\partial \Omega)$ whose boundaries have points withnon-unique blow-ups. Philosophically the examples arise from oscillating orrotating a blow-up limit by an infinite amount, but very slowly.

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