Regularity of the singular set in a two-phase problem for harmonic measure with Hölder data

Badger, Matthew; Engelstein, Max; Toro, Tatiana

doi:10.4171/rmi/1170

Cited by 7 publications

(3 citation statements)

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“…The proof of Theorem 1 uses tools from the theory of non-tangentially accessible domains (NTA) introduced by Jerison and Kenig [37], the monotonicity formula of Alt, Caffarelli, and Friedman [2], the theory of tangent measures introduced by Preiss [56], and the blow up techniques for harmonic measures at infinity for unbounded NTA domains due to Kenig and Toro [40,41]. For additional results along these lines see [11][12][13][14]29].…”

Section: Two Phase Casementioning

confidence: 99%

Geometric Measure Theory–Recent Applications

Toro¹

2019

Notices Amer. Math. Soc.

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Geometric Measure Theory (GMT) provides a framework to address questions in very different areas of mathematics, including calculus of variations, geometric analysis, potential theory, free boundary regularity, harmonic analysis, and theoretical computer science. Progress in different branches of GMT has led to the emergence of new challenges, making it a very vibrant area of research. In this note we will provide a historic background to some of the

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Section: Two Phase Casementioning

confidence: 99%

Geometric Measure Theory–Recent Applications

Toro¹

2019

Notices Amer. Math. Soc.

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“…For other works of more quantitative nature where one assumes Ω + , Ω − to be complementary NTA domains and either that ω − ∈ A ∞ (ω + ) or stronger conditions, see [KT3], [En], [AMT2], [PT], and [TT], for instance. See also [BET1] and [BET2] for other recent results dealing with the structure of the singular set of the boundary.…”

Section: Introductionmentioning

confidence: 99%

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

Tolsa

2024

Preprint

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Let $\Omega^+\subset\mathbb R^{n+1}$ be a vanishing Reifenberg flat domain such that $\Omega^+$ and $\Omega^-=\mathbb R^{n+1}\setminus\overline {\Omega^+}$ have joint big pieces of chord-arc subdomains and such that the outer unit normal to $\partial\Omega^+$ belongs to $VMO(\omega^+)$, where $\omega^\pm$ is the harmonic measure in $\Omega^\pm$. Up to now it was an open question if these conditions imply that $\log\dfrac{d\omega^-}{d\omega^+} \in VMO(\omega^+)$. 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 $\partial\Omega^+$ is constant $\omega^+$-a.e.\ in $\partial\Omega^+$.

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“…Under a stronger free boundary regularity hypothesis, the answer is affirmative. Following Engelstein [Eng16] and [BET20], we know that if log h ∈ C 0,α (∂Ω) for some α > 0 (Hölder continuous), then blow-ups are unique. Moreover, when log h ∈ C 0,α (∂Ω), the regular set Γ 1 is actually a C 1,α embedded submanifold and the singular set ∂Ω \ Γ 1 is (n − 3)-rectifiable in the sense of geometric measure theory (see e.g.…”

Section: Introductionmentioning

confidence: 99%