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manuscripta mathematica

, Volume 33, Issue 1, pp 81–98 | Cite as

Construction of a singular elliptic-harmonic measure

  • Luciano Modica
  • Stefano Mortola
Article

Abstract

The authors study an example, suggested by E. De Giorgi, of a second-order uniformly elliptic partial differential operator in divergence form with continuous coefficients on a smooth domain in the plane such that the associated “harmonic measure“ on the boundary is not absolutely continuous with respect to the ordinary surface measure.

Keywords

Differential Operator Number Theory Divergence Form Algebraic Geometry Topological Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    CAFFARELLI, L., FABES, E., KENIG, C.: Completely singular elliptic-harmonic measures. To appear on Indiana University Mathematics JournalGoogle Scholar
  2. [2]
    CAFFARELLI, L., FABES, E., MORTOLA, S., SALSA, S.: Boundary behavior of nonnegative solutions of elliptic operators in divergence form. To appear on Indiana University Mathematics JournalGoogle Scholar
  3. [3]
    DAHLBERG, B.E.J.: Estimates of harmonic measures. Arch. Rat. Mech. Analysis65, 275–288 (1977)Google Scholar
  4. [4]
    LITTMAN, W., STAMPACCHIA, G., WEINBERGER, H.F.: Regular points for elliptic equations with discontinuous coefficients. Ann. Sc. Norm. Sup. Pisa, Cl. Sci., Ser. III,XVII, 45–79 (1963)Google Scholar
  5. [5]
    MEYERS, N.G.: An LP-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Sc. Norm. Sup. Pisa, Cl. Sci., Ser. III,XVII, 189–206 (1963)Google Scholar
  6. [6]
    MOSER, J.: On Harnack's theorem for elliptic differential equations. Comm. Pure Appl. Math.XIV, 577–591 (1961)Google Scholar
  7. [7]
    PROTTER, M., WEINBERGER, H.F.: Maximum principles in differential equations. Englewood Cliffs N.J., Prentice-Hall Inc. 1967Google Scholar
  8. [8]
    ZYGMUND, A.: Trigonometric series. London-New York. Cambridge University Press 1959Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Luciano Modica
    • 1
  • Stefano Mortola
    • 1
  1. 1.Istituto di Matematica “Leonida Tonelli”Pisa(Italy)

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