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A SINGULAR INTERGRAL APPOACH TO A TWO PHASE FREE BOUNDARY PROBLEM.
- Source :
- Proceedings of the American Mathematical Society; Sep2016, Vol. 144 Issue 9, p3959-3973, 15p
- Publication Year :
- 2016
-
Abstract
- We present an alternative proof of a result of Kenig and Toro (2006), which states that if Ω ⊂ ℝ<superscript>n+1</superscript> 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<subscript>ext</subscript> 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<superscript>n</superscript> measure. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00029939
- Volume :
- 144
- Issue :
- 9
- Database :
- Complementary Index
- Journal :
- Proceedings of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 116340265
- Full Text :
- https://doi.org/10.1090/proc/13035