Mathematische Zeitschrift

, 264:15 | Cite as

Divergence form operators in Reifenberg flat domains

  • Emmanouil MilakisEmail author
  • Tatiana Toro


We study the boundary regularity of solutions of elliptic operators in divergence form with C 0,α coefficients or operators which are small perturbations of the Laplacian in non-smooth domains. We show that, as in the case of the Laplacian, there exists a close relationship between the regularity of the corresponding elliptic measure and the geometry of the domain.


Reifenberg flat domain Chord arc domain Elliptic measure 

Mathematics Subject Classification (2000)

35J25 31B05 


  1. 1.
    Caffarelli L., Fabes E., Mortola S., Salsa S.: Boundary behavior of nonnegative solutions of elliptic operators in divergence form. Indiana Univ. Math. J. 30(4), 621–640 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Caffarelli L., Kenig C.: Gradient estimates for variable coefficient parabolic equations and singular perturbation problems. Am. J. Math. 120(2), 391–439 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Coifman R., Fefferman C.: Weighted norm inequalities for maximal functions and singular integrals. Stud. Math. 51, 241–250 (1974)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Dahlberg B.: Estimates of harmonic measure. Arch. Ration. Mech. Anal. 65(3), 275–288 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dahlberg B.: On the absolute continuity of elliptic measure. Am. J. Math. 108, 1119–1138 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dahlberg B., Jerison D., Kenig C.: Area integral estimates for elliptic differential operators with nonsmooth coefficients. Ark. Mat. 22(1), 97–108 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Escauriaza L.: The L p Dirichlet problem for small perturbations of the Laplacian. Isr. J. Math. 94, 353–366 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Fefferman R.: A criterion for the absolute continuity of the harmonic measure associated with an elliptic operator. J. Am. Math. Soc. 2(1), 127–135 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fefferman R., Kenig C., Pipher J.: The theory of weights and the Dirichlet problem for elliptic equations. Ann. Math. (2) 134(1), 65–124 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gilbarg, D., Trudinger, N.: Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer, Berlin, pp. xiv+517 (2001)Google Scholar
  11. 11.
    Hardt R., Simon L.: Boundary regularity and embedded solutions for the oriented Plateau problem. Ann. Math. (2) 110(3), 439–486 (1979)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Jerison D., Kenig C.: The logarithm of the Poisson kernel of a C 1 domain has vanishing mean oscillation. Trans. Am. Math. Soc. 273(2), 781–794 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kenig, C.: Harmonic analysis techniques for second order elliptic boundary value problems. CBMS Regional Conference Series in Mathematics, vol. 83. AMS, Providence, pp. xii+146 (1994)Google Scholar
  14. 14.
    Kenig C., Pipher J.: The Dirichlet problem for elliptic equations with drift terms. Publ. Mat. 45(1), 199–217 (2001)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Kenig C., Toro T.: Harmonic measure on locally flat domains. Duke Math. J. 87(3), 509–551 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Korey, M.: Ideal Weights: Doubling and Absolute Continuity with Asymptotically Optimal Bounds, Ph.D. thesis, University of Chicago (1995)Google Scholar
  17. 17.
    Sarason D.: Functions of vanishing mean oscillation. Trans. Am. Math. Soc. 207, 391–405 (1975)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

Personalised recommendations