Advertisement

The Journal of Geometric Analysis

, Volume 28, Issue 2, pp 866–908 | Cite as

BMO Solvability and \(A_{\infty }\) Condition of the Elliptic Measures in Uniform Domains

Article

Abstract

We consider the Dirichlet boundary value problem for divergence form elliptic operators with bounded measurable coefficients. We prove that for uniform domains with Ahlfors regular boundary, the BMO solvability of such problems is equivalent to a quantitative absolute continuity of the elliptic measure with respect to the surface measure, i.e., \(\omega _L\in A_{\infty }(\sigma )\). This generalizes a previous result on Lipschitz domains by Dindos, Kenig, and Pipher (see Dindos et al. in J Geom Anal 21:78–95, 2011).

Keywords

Harmonic measure Uniform domain \(A_{\infty }\) Muckenhoupt weight BMO solvability Carleson measure 

Mathematics Subject Classification

35J25 31B35 42B37 

Notes

Acknowledgements

The author was partially supported by NSF DMS Grants 1361823 and 1500098. The author wants to thank her advisor Prof. Tatiana Toro for introducing her to this area and the enormous support given by her during the work on this paper. The author also wants to thank Prof. Hart Smith for his support, and thank the referee for the careful reading and helpful suggestions.

References

  1. 1.
    Aikawa, H.: Norm estimate of Green operator, perturbation of Green function and integrability of superharmonic functions. Math. Ann. 312, 289–318 (1998)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aikawa, H.: Boundary Harnack principle and Martin boundary for a uniform domain. J. Math. Soc. Jpn. 53(1), 119–145 (2001)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Aikawa, H.: Potential-theoretic characterizations of nonsmooth domains. Bull. Lond. Math. Soc. 36, 469–482 (2004)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ancona, A.: On strong barriers and an inequality of Hardy for domains in \(\mathbb{R}^n\). J. Lond. Math. Soc. 34, 274–290 (1986)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Azzam, J., Hofmann, S., Martell, J.M., Mayboroda, S., Mourgoglou, M., Tolsa, X., Volberg, A.: Rectifiability of harmonic measure. Preprint, arXiv:1509.06294
  6. 6.
    Azzam, J., Hofmann, S., Martell, J.M., Nyström, K., Toro, T.: A new characterization of chord-arc domains. arXiv:1406.2743, to appear in J. Eur. Math. Soc
  7. 7.
    Carleson, L.: On the existence of boundary values of harmonic functions of several variables. Ark. Math. 4, 339–393 (1962)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Caffarelli, L., Fabes, E., Mortola, S., Salsa, S.: Boundary behavior of non-negative solutions of elliptic operators in divergence form. Indiana Univ. Math. J. 30, 621–640 (1981)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Coifman, R.R., Rochberg, R.: Another characterization of BMO. Proc. Am. Math. Soc. 79, 249–254 (1980)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dahlberg, B.E.J.: On estimates for harmonic measure. Arch. Ration. Mech. Anal. 65, 272–288 (1977)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dahlberg, B.E.J.: On the absolute continuity of elliptic measures. Am. J. Math. 108(5), 1119–1138 (1986)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    David, G., Jerison, D.: Lipschitz approximation to hypersurfaces, harmonic measure, and singular integrals. Indiana Univ. Math. J. 39(3), 831–845 (1990)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dindos, M., Kenig, C., Pipher, J.: BMO solvability and the \(A_{\infty }\) condition for elliptic operators. J. Geom. Anal. 21(1), 78–95 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)MATHGoogle Scholar
  15. 15.
    Fabes, E., Kenig, C., Neri, U.: Carleson measures, \(H^1\) duality and weighted BMO in non-smooth domains. Indiana J. Math. 30(4), 547–581 (1981)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Fabes, E., Neri, U.: Dirichlet problem in Lipschitz domains with BMO data. Proc. Am. Math. Soc. 78, 33–39 (1980)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Fefferman, R.: A criterion for the absolute continuity of the harmonic measure associated with an elliptic operator. J. AMS, vol. 2, Number 1, 134, 65–124 (1989)Google Scholar
  18. 18.
    Fefferman, R., Kenig, C., Pipher, J.: The theory of weights and the Dirichlet problem for elliptic equations. Ann. Math. Second Ser. 134(1), 65–124 (1991)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Fabes, E., Neri, U.: Dirichlet problem in Lipschitz domains with BMO data. Proc. Am. Math. Soc. 78, 165–186 (1980)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Fefferman, C., Stein, E.M.: \(H^p\) spaces of several variables. Acta Math. 129, 137–193 (1972)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)MATHGoogle Scholar
  22. 22.
    Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, New York (1993)MATHGoogle Scholar
  23. 23.
    Hofmann, S., Le, P.: BMO solvability and absolute continuity of harmonic measure. arXiv:1607.00418v1
  24. 24.
    Hofmann, S., Martell, J.M.: Uniform rectifiability and harmonic measure I: uniform rectifiability implies Poisson kernels in \(L^p\). Ann. Sci. Ecole Norm. Sup. 47(3), 577–654 (2014)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Hofmann, S., Martell, J.M., Mayboroda, S.: Uniform rectifiability, Carleson measure estimates, and approximation of harmonic functions. To appear in Duke Math. J. arXiv:1408.1447
  26. 26.
    Hofmann, S., Martell, J.M., Uriarte-Tuero, I.: Uniform rectifiability and harmonic measure II: Poisson kernels in \(L^p\) imply uniform rectifiability. To appear in Duke Math. J. arXiv:1202.3860v2
  27. 27.
    Hofmann, S., Martell, J.M., Toro, T.: Elliptic Operators on Non-smooth Domains. Book in preparationGoogle Scholar
  28. 28.
    Hunt, R., Wheeden, R.: Positive harmonic functions on Lipschitz domains. Trans. Am. Math. Soc. 147, 507–527 (1970)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Jones, P.W.: A geometric localization theorem. Adv. Math. 46, 71–79 (1982)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Jerison, D., Kenig, C.: Boundary behavior of harmonic functions in non-tangentially accessible domains. Adv. Math. 46, 80–147 (1982)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Kenig, C.: Harmonic analysis techniques for second order elliptic boundary value problems. In: Proceedings of the CBMS regional conference series in mathematics, 83. AMS Providence, RI (1994)Google Scholar
  32. 32.
    Kenig, C., Kirchheim, B., Pipher, J., Toro, T.: Square functions and the \(A_{\infty }\) property of elliptic measures. J. Geom. Anal. 26, 2383 (2016). doi: 10.1007/s12220-015-9630-6 MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Kenig, C., Koch, H., Pipher, J., Toro, T.: A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations. Adv. Math. 153(2), 231–298 (2000). doi: 10.1006/aima.1999.1899 MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Littman, W., Stampacchia, G., Weinberger, H.F.: Regular points for elliptic equations with discontinuous coefficients. Annali della Scuola Normale Superiore di Pisa—Classe di Scienze 17(1–2), 43–77 (1963)MathSciNetMATHGoogle Scholar
  35. 35.
    Milakis, E., Pipher, J., Toro, T.: Harmonic analysis on chord arc domain. J. Geom. Anal. 23, 2091–2157 (2013)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Semmes, S.: Analysis vs. geometry on a class of rectifiable hypersurfaces in \({\mathbb{R}}^{n}\). Indiana Univ. Math. J. 39(4), 10051035 (1990)MathSciNetGoogle Scholar
  37. 37.
    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, 1st edn. Princeton University Press, Princeton (1993)MATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

Personalised recommendations