# A singular integral approach to a two phase free boundary problem

@inproceedings{Bortz2015ASI, title={A singular integral approach to a two phase free boundary problem}, author={Simon Bortz and Steve Hofmann}, year={2015} }

We present an alternative proof of a result of Kenig and Toro, which states that if $\Omega \subset \mathbb{R}^{n+1}$ is a two sided NTA domain, with Ahlfors-David regular boundary, and the $\log$ of the Poisson kernel associated to $\Omega$ as well as the $\log$ of the Poisson kernel associated to ${\Omega_{\rm ext}}$ are in VMO, then the outer unit normal $\nu$ is in VMO . Our proof exploits the usual jump relation formula for the non-tangential limit of the gradient of the single layer…

## 7 Citations

The one-phase problem for harmonic measure in two-sided NTA domains

- Mathematics
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We show that if $\Omega\subset\mathbb R^3$ is a two-sided NTA domain with AD-regular boundary such that the logarithm of the Poisson kernel belongs to $\textrm{VMO}(\sigma)$, where $\sigma$ is the…

The two-phase problem for harmonic measure in VMO

- MathematicsCalculus of Variations and Partial Differential Equations
- 2020

Let $$\Omega ^+\subset {\mathbb {R}}^{n+1}$$ Ω + ⊂ R n + 1 be an NTA domain and let $$\Omega ^-= {\mathbb {R}}^{n+1}\setminus \overline{\Omega ^+}$$ Ω - = R n + 1 \ Ω + ¯ be an NTA domain as well.…

Structure of sets which are well approximated by zero sets of harmonic polynomials

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The zero sets of harmonic polynomials play a crucial role in the study of the free boundary regularity problem for harmonic measure. In order to understand the fine structure of these free boundaries…

Two phase free boundary problem for Poisson kernels

- MathematicsIndiana University Mathematics Journal
- 2022

We provide a potential theoretic characterization of vanishing chord-arc domains under minimal assumptions. In particular we show that, in the appropriate class of domains, the oscillation of the…

$L^2$-boundedness of gradients of single layer potentials for elliptic operators with coefficients of Dini mean oscillation-type

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- 2021

We consider a uniformly elliptic operator LA in divergence form associated with an (n+1)×(n+1)-matrix A with real, merely bounded, and possibly non-symmetric coefficients. If ωA(r) = sup x∈Rn+1 − ∫

Reifenberg Flatness and Oscillation of the Unit Normal Vector

- Mathematics
- 2017

We show (under mild topological assumptions) that small oscillation of the unit normal vector implies Reifenberg flatness. We then apply this observation to the study of chord-arc domains and to a…

Mutual Absolute Continuity of Interior and Exterior Harmonic Measure Implies Rectifiability

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We show that, for disjoint domains in the euclidean space whose boundaries satisfy a nondegeneracy condition, mutual absolute continuity of their harmonic measures implies absolute continuity with…

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