TY - JOUR
T1 - Slowly Vanishing Mean Oscillations
T2 - Non-uniqueness of Blow-ups in a Two-phase Free Boundary Problem
AU - Badger, Matthew
AU - Engelstein, Max
AU - Toro, Tatiana
N1 - Publisher Copyright:
© 2023, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.
PY - 2023
Y1 - 2023
N2 - 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 log h∈ C,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.
AB - 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 log h∈ C,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.
KW - Harmonic measure
KW - Two-phase free boundary problems
KW - Uniqueness of blow-ups
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U2 - 10.1007/s10013-023-00668-6
DO - 10.1007/s10013-023-00668-6
M3 - Article
AN - SCOPUS:85178886770
SN - 2305-221X
JO - Vietnam Journal of Mathematics
JF - Vietnam Journal of Mathematics
ER -
