## Abstract

We present an alternative proof of a result of Kenig and Toro (2006), which states that if Ω ⊂ ℝ^{n+1} is a 2-sided NTA domain, with Ahlfors- David regular boundary, and the log of the Poisson kernel associated to Ω as well as the log of the Poisson kernel associated to Ω_{ext} are in VMO, then the outer unit normal ν is in VMO. Our proof exploits the usual jump relation formula for the non-tangential limit of the gradient of the single layer potential. We are also able to relax the assumptions of Kenig and Toro in the case that the pole for the Poisson kernel is finite: in this case, we assume only that ∂Ω is uniformly rectifiable, and that ∂Ω coincides with the measure theoretic boundary of Ω a.e. with respect to Hausdorff H^{n} measure.

Original language | English (US) |
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Pages (from-to) | 3959-3973 |

Number of pages | 15 |

Journal | Proceedings of the American Mathematical Society |

Volume | 144 |

Issue number | 9 |

DOIs | |

State | Published - 2016 |

### Bibliographical note

Funding Information:NSF grant DMS-1361701.

## Keywords

- Free boundary problems
- Layer potentials
- Poisson kernels
- Singular integrals
- VMO