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arXiv · 2210.17531

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

Matthew Badger·Max Engelstein·Tatiana Toro
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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.

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Matthew Badger, Max Engelstein, Tatiana Toro. 2022-10-31. Slowly vanishing mean oscillations: non-uniqueness of blow-ups in a two-phase free boundary problem. https://arxiv.org/abs/2210.17531

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