A counterexample regarding a two-phase problem for harmonic measure in VMO
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| Format: | Preprint |
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2024
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| _version_ | 1866916370395955200 |
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| author | Tolsa, Xavier |
| author_facet | Tolsa, Xavier |
| contents | Let $Ω^+\subset\mathbb R^{n+1}$ be a vanishing Reifenberg flat domain such that $Ω^+$ and $Ω^-=\mathbb R^{n+1}\setminus\overline {Ω^+}$ have joint big pieces of chord-arc subdomains and the outer unit normal to $Ω^+$ belongs to $VMO(ω^+)$, where $ω^\pm$ is the harmonic measure of $Ω^\pm$. Up to now it was an open question if these conditions imply that $\log\dfrac{dω^-}{dω^+} \in VMO(ω^+)$. In this paper we answer this question in the negative by constructing an appropriate counterexample in $\mathbb R^2$, with the additional property that the outer unit normal to $Ω^+$ is constant $ω^+$-a.e. in $\partialΩ^+$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_14568 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A counterexample regarding a two-phase problem for harmonic measure in VMO Tolsa, Xavier Analysis of PDEs Classical Analysis and ODEs 31B15, 31B20, 28A75 Let $Ω^+\subset\mathbb R^{n+1}$ be a vanishing Reifenberg flat domain such that $Ω^+$ and $Ω^-=\mathbb R^{n+1}\setminus\overline {Ω^+}$ have joint big pieces of chord-arc subdomains and the outer unit normal to $Ω^+$ belongs to $VMO(ω^+)$, where $ω^\pm$ is the harmonic measure of $Ω^\pm$. Up to now it was an open question if these conditions imply that $\log\dfrac{dω^-}{dω^+} \in VMO(ω^+)$. In this paper we answer this question in the negative by constructing an appropriate counterexample in $\mathbb R^2$, with the additional property that the outer unit normal to $Ω^+$ is constant $ω^+$-a.e. in $\partialΩ^+$. |
| title | A counterexample regarding a two-phase problem for harmonic measure in VMO |
| topic | Analysis of PDEs Classical Analysis and ODEs 31B15, 31B20, 28A75 |
| url | https://arxiv.org/abs/2401.14568 |