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=dω^-/dω⁺ of harmonic measures on complementary NTA domains controls the geometry of their common boundary. It is now known that logh∈C^0,α(∂Ω) 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 logh∈C(∂Ω) 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.