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A singular integral approach to a two phase free boundary problem


Authors: Simon Bortz and Steve Hofmann
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
Published electronically: March 17, 2016
MathSciNet review: 3513552
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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 _{\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.


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Additional Information

Simon Bortz
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: sabh8f@mail.missouri.edu

Steve Hofmann
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: hofmanns@missouri.edu

DOI: https://doi.org/10.1090/proc/13035
Keywords: Singular integrals, layer potentials, free boundary problems, Poisson kernels, VMO
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
Article copyright: © Copyright 2016 American Mathematical Society

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