A counterexample regarding a two-phase problem for harmonic measure in VMO

2024/01/25 by Xavier Tolsa, Tolsa, Xavier
Computer Science · Mathematics · #28A75 #31B15 #31B20 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2401.14568

openalex publication_date 2024/01/25 · openalex created_date 2024/01/30 · openalex updated_date 2026/07/30

Abstract

Let Ω⁺⊂\mathbb Rⁿ⁺¹ be a vanishing Reifenberg flat domain such that Ω⁺ and Ω^-=\mathbb Rⁿ⁺¹∖ Ω⁺ have joint big pieces of chord-arc subdomains and the outer unit normal to Ω⁺ belongs to VMO(ω⁺), where ω^± is the harmonic measure of Ω^±. Up to now it was an open question if these conditions imply that log\dfracdω^-dω⁺ ∈ VMO(ω⁺). In this paper we answer this question in the negative by constructing an appropriate counterexample in \mathbb R², with the additional property that the outer unit normal to Ω⁺ is constant ω⁺-a.e. in ∂Ω⁺.

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