A singular integral approach to a two phase free boundary problem

  title={A singular integral approach to a two phase free boundary problem},
  author={Simon Bortz and Steve Hofmann},
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… 
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