A Counterexample Regarding a Two-phase Problem for Harmonic Measure in VMO
Let Ω+⊂Rn+1 be a vanishing Reifenberg flat domain such that Ω+ and Ω- = Rn+1\Ω+¯ have joint big pieces of chord-arc subdomains and such that the outer unit normal to ∂Ω+ belongs to VMO(ω+), where ω± is the harmonic measure in Ω±. Up to now it was an open question if these conditions imply that logdω...
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2025 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:311855 |
| Acceso en línea: | https://ddd.uab.cat/record/311855 https://dx.doi.org/urn:doi:10.1007/s11118-025-10214-3 |
| Access Level: | acceso abierto |
| Palabra clave: | Harmonic measure Two-phase problem Reifenberg flat domains |
| Sumario: | Let Ω+⊂Rn+1 be a vanishing Reifenberg flat domain such that Ω+ and Ω- = Rn+1\Ω+¯ have joint big pieces of chord-arc subdomains and such that the outer unit normal to ∂Ω+ belongs to VMO(ω+), where ω± is the harmonic measure in Ω±. Up to now it was an open question if these conditions imply that logdω-dω+∈VMO(ω+). In this paper we answer this question in the negative by constructing an appropriate counterexample in R2, with the additional property that the outer unit normal to ∂Ω+ is constant ω+-a.e. in ∂Ω+. |
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