We show that if ω ⊂ R3 is a two-sided NTA domain with AD-regular boundary such that the logarithm of the Poisson kernel belongs to VMO(σ), where σ is the surface measure of ω, then the outer unit normal to ∂ω belongs to VMO(σ) too. The analogous result fails for dimensions larger than 3. This answers a question posed by Kenig and Toro and also by Bortz and Hofmann.
- Free boundary problem
- Harmonic measure
- Nontangentially accessible (NTA) domains
- Poisson kernel