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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
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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  • [1] H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105-144. MR 618549 (83a:49011)
  • [2] J. Bourgain, On the Hausdorff dimension of harmonic measure in higher dimension, Invent. Math. 87 (1987), no. 3, 477-483. MR 874032 (88b:31004),
  • [3] L. Caffarelli, E. Fabes, S. Mortola, and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J. 30 (1981), no. 4, 621-640. MR 620271 (83c:35040),
  • [4] G. David and D. Jerison, Lipschitz approximation to hypersurfaces, harmonic measure, and singular integrals, Indiana Univ. Math. J. 39 (1990), no. 3, 831-845. MR 1078740 (92b:42021),
  • [5] G. David and S. Semmes, Singular integrals and rectifiable sets in $ {\bf R}^n$: Beyond Lipschitz graphs, Astérisque 193 (1991), 152 (English, with French summary). MR 1113517 (92j:42016)
  • [6] Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, Mathematical Surveys and Monographs, vol. 38, American Mathematical Society, Providence, RI, 1993. MR 1251061 (94i:28003)
  • [7] Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660 (93f:28001)
  • [8] José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149 (87d:42023)
  • [9] Steve Hofmann, Marius Mitrea, and Michael Taylor, Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains, Int. Math. Res. Not. IMRN 14 (2010), 2567-2865. MR 2669659 (2011h:42021),
  • [10] David Jerison, Regularity of the Poisson kernel and free boundary problems, Colloq. Math. 60/61 (1990), no. 2, 547-568. MR 1096396 (92b:35177)
  • [11] David S. Jerison and Carlos E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math. 46 (1982), no. 1, 80-147. MR 676988 (84d:31005b),
  • [12] Oliver Dimon Kellogg, Foundations of potential theory, Reprint from the first edition of 1929. Die Grundlehren der Mathematischen Wissenschaften, Band 31, Springer-Verlag, Berlin-New York, 1967. MR 0222317 (36 #5369)
  • [13] Carlos E. Kenig and Tatiana Toro, Harmonic measure on locally flat domains, Duke Math. J. 87 (1997), no. 3, 509-551. MR 1446617 (98k:31010),
  • [14] Carlos E. Kenig and Tatiana Toro, Free boundary regularity for harmonic measures and Poisson kernels, Ann. of Math. (2) 150 (1999), no. 2, 369-454. MR 1726699 (2001d:31004),
  • [15] Carlos E. Kenig and Tatiana Toro, Poisson kernel characterization of Reifenberg flat chord arc domains, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 3, 323-401 (English, with English and French summaries). MR 1977823 (2004d:31010),
  • [16] Carlos Kenig and Tatiana Toro, Free boundary regularity below the continuous threshold: 2-phase problems, J. Reine Angew. Math. 596 (2006), 1-44. MR 2254803 (2007k:35526),
  • [17] Oldřich Kowalski and David Preiss, Besicovitch-type properties of measures and submanifolds, J. Reine Angew. Math. 379 (1987), 115-151. MR 903637 (88h:49070)
  • [18] Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890 (96h:28006)
  • [19] Pertti Mattila, Mark S. Melnikov, and Joan Verdera, The Cauchy integral, analytic capacity, and uniform rectifiability, Ann. of Math. (2) 144 (1996), no. 1, 127-136. MR 1405945 (97k:31004),
  • [20] Fedor Nazarov, Xavier Tolsa, and Alexander Volberg, On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1, Acta Math. 213 (2014), no. 2, 237-321. MR 3286036,

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Additional Information

Simon Bortz
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Steve Hofmann
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

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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