A singular integral approach to a two phase free boundary problem
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- by Simon Bortz and Steve Hofmann PDF
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Abstract:
We present an alternative proof of a result of Kenig and Toro (2006), which states that if $\Omega \subset \mathbb {R}^{n+1}$ is a 2-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 ${\Omega _\textrm {ext}}$ 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.References
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Additional Information
- Simon Bortz
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 1166754
- ORCID: 0000-0001-7955-3035
- Email: sabh8f@mail.missouri.edu
- Steve Hofmann
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 251819
- ORCID: 0000-0003-1110-6970
- Email: hofmanns@missouri.edu
- Received by editor(s): May 19, 2015
- Received by editor(s) in revised form: November 17, 2015
- Published electronically: March 17, 2016
- Additional Notes: The authors were supported by NSF grant DMS-1361701.
- Communicated by: Svitlana Mayboroda
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 3959-3973
- MSC (2010): Primary 42B20, 31B05, 31B25, 35J08, 35J25
- DOI: https://doi.org/10.1090/proc/13035
- MathSciNet review: 3513552