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

2022/10/31 by Matthew Badger, Badger, Matthew, Max Engelstein +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2210.17531

openalex publication_date 2022/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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^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.

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