The two-phase problem for harmonic measure in VMO and the chord-arc condition
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866916168660418560 |
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| author | Tolsa, Xavier Toro, Tatiana |
| author_facet | Tolsa, Xavier Toro, Tatiana |
| contents | Let $Ω^+\subset\mathbb R^{n+1}$ be a bounded $δ$-Reifenberg flat domain, with $δ>0$ small enough, possibly with locally infinite surface measure. Assume also that $Ω^-= \mathbb R^{n+1}\setminus \overline{Ω^+}$ is an NTA domain as well and denote by $ω^+$ and $ω^-$ the respective harmonic measures of $Ω^+$ and $Ω^-$ with poles $p^\pm\inΩ^\pm$. In this paper we show that the condition that $\log\dfrac{dω^-}{dω^+} \in VMO(ω^+)$ is equivalent to $Ω^+$ being a chord-arc domain with inner normal belonging to $VMO(H^n|_{\partialΩ^+})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_14346 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The two-phase problem for harmonic measure in VMO and the chord-arc condition Tolsa, Xavier Toro, Tatiana Analysis of PDEs Classical Analysis and ODEs 31A15, 28A75, 35R35 Let $Ω^+\subset\mathbb R^{n+1}$ be a bounded $δ$-Reifenberg flat domain, with $δ>0$ small enough, possibly with locally infinite surface measure. Assume also that $Ω^-= \mathbb R^{n+1}\setminus \overline{Ω^+}$ is an NTA domain as well and denote by $ω^+$ and $ω^-$ the respective harmonic measures of $Ω^+$ and $Ω^-$ with poles $p^\pm\inΩ^\pm$. In this paper we show that the condition that $\log\dfrac{dω^-}{dω^+} \in VMO(ω^+)$ is equivalent to $Ω^+$ being a chord-arc domain with inner normal belonging to $VMO(H^n|_{\partialΩ^+})$. |
| title | The two-phase problem for harmonic measure in VMO and the chord-arc condition |
| topic | Analysis of PDEs Classical Analysis and ODEs 31A15, 28A75, 35R35 |
| url | https://arxiv.org/abs/2209.14346 |