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Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries


Authors: Murat Akman, Matthew Badger, Steve Hofmann and José María Martell
Journal: Trans. Amer. Math. Soc. 369 (2017), 5711-5745
MSC (2010): Primary 28A75, 28A78, 31A15, 31B05, 35J25, 42B37, 49Q15
DOI: https://doi.org/10.1090/tran/6927
Published electronically: April 24, 2017
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Abstract: Let $ \Omega \subset \mathbb{R}^{n+1}$, $ n\geq 2$, be a 1-sided NTA domain (also known as a uniform domain), i.e., a domain which satisfies interior corkscrew and Harnack chain conditions, and assume that $ \partial \Omega $ is $ n$-dimensional Ahlfors-David regular. We characterize the rectifiability of $ \partial \Omega $ in terms of the absolute continuity of surface measure with respect to harmonic measure. We also show that these are equivalent to the fact that $ \partial \Omega $ can be covered $ \mathcal {H}^n$-a.e. by a countable union of portions of boundaries of bounded chord-arc subdomains of $ \Omega $ and to the fact that $ \partial \Omega $ possesses exterior corkscrew points in a qualitative way $ \mathcal {H}^n$-a.e. Our methods apply to harmonic measure and also to elliptic measures associated with real symmetric second order divergence form elliptic operators with locally Lipschitz coefficients whose derivatives satisfy a natural qualitative Carleson condition.


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  • [1] David R. Adams and Lars Inge Hedberg, Function spaces and potential theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 314, Springer-Verlag, Berlin, 1996. MR 1411441
  • [2] Hiroaki Aikawa and Kentaro Hirata, Doubling conditions for harmonic measure in John domains, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 2, 429-445 (English, with English and French summaries). MR 2410379
  • [3] Jonas Azzam, Steve Hofmann, José María Martell, Svitlana Mayboroda, Mihalis Mourgoglou, Xavier Tolsa, and Alexander Volberg, Rectifiability of harmonic measure, Geom. Funct. Anal. 26 (2016), no. 3, 703-728. MR 3540451, https://doi.org/10.1007/s00039-016-0371-x
  • [4] Jonas Azzam, Steve Hofmann, José María Martell, Kaj Nyström, and Tatiana Toro, A new characterization of chord-arc domains, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 4, 967-981. MR 3626548, https://doi.org/10.4171/JEMS/685
  • [5] J. Azzam, M. Mourgoglou, and X. Tolsa, Singular sets for harmonic measure on locally flat domains with locally finite surface measure, to appear in Int. Math. Res. Not., arXiv:1501.07585.
  • [6] J. Azzam, M. Mourgoglou, and X. Tolsa, Rectifiability of harmonic measure in domains with porous boundaries, preprint, arXiv:1505.06088.
  • [7] Matthew Badger, Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited, Math. Z. 270 (2012), no. 1-2, 241-262. MR 2875832, https://doi.org/10.1007/s00209-010-0795-1
  • [8] Christopher J. Bishop and Peter W. Jones, Harmonic measure and arclength, Ann. of Math. (2) 132 (1990), no. 3, 511-547. MR 1078268, https://doi.org/10.2307/1971428
  • [9] Simon Bortz and Steve Hofmann, Harmonic measure and approximation of uniformly rectifiable sets, Rev. Mat. Iberoam. 33 (2017), no. 1, 351-373. MR 3615455, https://doi.org/10.4171/RMI/940
  • [10] J. Bourgain, On the Hausdorff dimension of harmonic measure in higher dimension, Invent. Math. 87 (1987), no. 3, 477-483. MR 874032, https://doi.org/10.1007/BF01389238
  • [11] Michael Christ, A $ T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, 601-628. MR 1096400
  • [12] Björn E. J. Dahlberg, On the absolute continuity of elliptic measures, Amer. J. Math. 108 (1986), no. 5, 1119-1138. MR 859772, https://doi.org/10.2307/2374598
  • [13] Björn E. J. Dahlberg, David S. Jerison, and Carlos E. Kenig, Area integral estimates for elliptic differential operators with nonsmooth coefficients, Ark. Mat. 22 (1984), no. 1, 97-108. MR 735881, https://doi.org/10.1007/BF02384374
  • [14] 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, https://doi.org/10.1512/iumj.1990.39.39040
  • [15] G. David and S. Semmes, Singular integrals and rectifiable sets in $ {\bf R}^n$: Beyond Lipschitz graphs, Astérisque 193 (1991), 152 (English, with French summary). MR 1113517
  • [16] Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, Mathematical Surveys and Monographs, vol. 38, American Mathematical Society, Providence, RI, 1993. MR 1251061
  • [17] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
  • [18] Michael Grüter and Kjell-Ove Widman, The Green function for uniformly elliptic equations, Manuscripta Math. 37 (1982), no. 3, 303-342. MR 657523, https://doi.org/10.1007/BF01166225
  • [19] Juha Heinonen, Tero Kilpeläinen, and Olli Martio, Nonlinear potential theory of degenerate elliptic equations, Dover Publications, Inc., Mineola, NY, 2006. Unabridged republication of the 1993 original. MR 2305115
  • [20] S. Hofmann. P. Le, J. M. Martell, and K. Nyström, The weak-$ A_\infty $ property of harmonic and $ p$-harmonic measures implies uniform rectifiability, to appear in Anal. PDE, arXiv:1511.09270.
  • [21] Steve Hofmann and José María Martell, Uniform rectifiability and harmonic measure I: Uniform rectifiability implies Poisson kernels in $ L^p$, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 3, 577-654 (English, with English and French summaries). MR 3239100
  • [22] S. Hofmann and J. M. Martell, Uniform Rectifiability and harmonic measure IV: Ahlfors regularity plus Poisson kernels in $ L^p$ implies uniform rectifiability, preprint, arXiv:1505.06499.
  • [23] S. Hofmann, J. M. Martell, S. Mayboroda, X. Tolsa, and A. Volberg, Absolute continuity between the surface measure and harmonic measure implies rectifiability, preprint, arXiv:1507.04409.
  • [24] S. Hofmann, J. M. Martell, and T. Toro, $ A_\infty $ implies NTA for a class of variable coefficient elliptic operators, preprint, arXiv:1611.09561.
  • [25] S. Hofmann, J. M. Martell, and T. Toro, General divergence form elliptic operators on domains with ADR boundaries, and on $ 1$-sided NTA domains, in preparation.
  • [26] Steve Hofmann, José María Martell, and Ignacio Uriarte-Tuero, Uniform rectifiability and harmonic measure, II: Poisson kernels in $ L^p$ imply uniform rectifiability, Duke Math. J. 163 (2014), no. 8, 1601-1654. MR 3210969, https://doi.org/10.1215/00127094-2713809
  • [27] Steve Hofmann, Dorina Mitrea, Marius Mitrea, and Andrew J. Morris, $ L^p$-square function estimates on spaces of homogeneous type and on uniformly rectifiable sets, Mem. Amer. Math. Soc. 245 (2017), no. 1159, v+108. MR 3589162, https://doi.org/10.1090/memo/1159
  • [28] 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, https://doi.org/10.1016/0001-8708(82)90055-X
  • [29] Carlos E. Kenig and Jill Pipher, The Dirichlet problem for elliptic equations with drift terms, Publ. Mat. 45 (2001), no. 1, 199-217. MR 1829584, https://doi.org/10.5565/PUBLMAT_45101_09
  • [30] M. Lavrentev, Boundary problems in the theory of univalent functions, Amer. Math. Soc. Transl. (2) 32 (1963), 1-35. MR 0155970
  • [31] John L. Lewis and Kaj Nyström, Regularity and free boundary regularity for the $ p$-Laplace operator in Reifenberg flat and Ahlfors regular domains, J. Amer. Math. Soc. 25 (2012), no. 3, 827-862. MR 2904575, https://doi.org/10.1090/S0894-0347-2011-00726-1
  • [32] Pertti Mattila, Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. MR 1333890
  • [33] M. Mourgoglou, Uniform domains with rectifiable boundaries and harmonic measure, preprint, arXiv:1505.06167.
  • [34] 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, https://doi.org/10.1007/s11511-014-0120-7
  • [35] Fedor Nazarov, Xavier Tolsa, and Alexander Volberg, The Riesz transform, rectifiability, and removability for Lipschitz harmonic functions, Publ. Mat. 58 (2014), no. 2, 517-532. MR 3264510
  • [36] Stephen Semmes, Analysis vs. geometry on a class of rectifiable hypersurfaces in $ {\bf R}^n$, Indiana Univ. Math. J. 39 (1990), no. 4, 1005-1035. MR 1087183, https://doi.org/10.1512/iumj.1990.39.39048
  • [37] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • [38] Jang-Mei Wu, On singularity of harmonic measure in space, Pacific J. Math. 121 (1986), no. 2, 485-496. MR 819202
  • [39] William P. Ziemer, Some remarks on harmonic measure in space, Pacific J. Math. 55 (1974), 629-637. MR 0427657

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Additional Information

Murat Akman
Affiliation: Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera, 13-15, E-28049 Madrid, Spain
Address at time of publication: Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, California 94720
Email: makman@msri.org

Matthew Badger
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
Email: matthew.badger@uconn.edu

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

José María Martell
Affiliation: Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera, 13-15, E-28049 Madrid, Spain
Email: chema.martell@icmat.es

DOI: https://doi.org/10.1090/tran/6927
Keywords: NTA domains, 1-sided NTA domains, uniform domains, Ahlfors-David regular sets, rectifiability, harmonic measure, elliptic measure, surface measure, linearly approximability, elliptic operators
Received by editor(s): July 9, 2015
Received by editor(s) in revised form: January 7, 2016
Published electronically: April 24, 2017
Additional Notes: The first and last authors have been supported in part by the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554), and they acknowledge that the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC agreement no. 615112 HAPDEGMT. The second author was partially supported by an NSF postdoctoral fellowship, DMS 1203497, and by NSF grant DMS 1500382. The third author was partially supported by NSF grant DMS 1361701.
Article copyright: © Copyright 2017 American Mathematical Society

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