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 ω - / d ω + of harmonic measures on complementary NTA domains controls the geometry of their common boundary. It is now known that log h ∈ C 0 , α ( ∂ Ω ) implies that pointwise...

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Bibliographic Details
Published in:Vietnam journal of mathematics 2024-07, Vol.52 (3), p.615-625
Main Authors: Badger, Matthew, Engelstein, Max, Toro, Tatiana
Format: Article
Language:English
Subjects:
Free boundaries
Mathematics
Mathematics and Statistics
Original Article
Polynomials
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Summary:In Kenig and Toro’s two-phase free boundary problem, one studies how the regularity of the Radon–Nikodym derivative h = d ω - / d ω + of harmonic measures on complementary NTA domains controls the geometry of their common boundary. It is now known that log h ∈ 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 log h ∈ 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.
ISSN:2305-221X
2305-2228
DOI:10.1007/s10013-023-00668-6