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

Abstract

In Kenig and Toro's two-phase free boundary problem, one studies how the regularity of the Radon-Nikodym derivative $h=d\ddot{I}{‰}^{-}/d\ddot{I}{‰}^{+}$ of harmonic measures on complementary NTA domains controls the geometry of their common boundary. It is now known that $\log h\in C^{0,\hat{I}\pm }(\partial \hat{I}\text{c◯})$ 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 $\log h\in C(\partial \hat{I}\text{c◯})$ 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.

13 pages, 9 figures. Dedicated to Carlos Kenig's 70th birthday. Comments welcome