A Singular Integral approach to a Two Phase Free Boundary Problem

TL;DR

This paper offers a new proof for a regularity result of the boundary normal in two-sided NTA domains with Ahlfors-David regular boundaries, using a singular integral approach and relaxing previous assumptions.

Contribution

It provides an alternative proof of a boundary regularity result and relaxes assumptions when the Poisson kernel pole is finite, using a singular integral method.

Findings

Outer unit normal is in VMO under given conditions.

New proof exploits jump relation formula for single layer potential.

Relaxed assumptions for finite pole case, requiring only uniform rectifiability.

Abstract

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 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 $\partial\Omega$ is uniformly rectifiable, and that $\partial\Omega$ coincides with the measure theoretic boundary of $\Omega$ a.e. with respect to Hausdorff $H^n$ measure.