Skip to main content
Log in

Removable sets for Hölder continuous solutions of quasilinear elliptic equations with lower order terms

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We consider quasi-linear second order elliptic differential equations with lower order terms and study removable sets for Hölder continuous solutions of the equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Adams, D.R., Hedberg, L.I.: Function spaces and potential theory. Springer- Verlag, Berlin (1995)

    MATH  Google Scholar 

  2. Carleson, L.: Selected problems on exceptional sets. Van Nostrand Mathematical Studies, New Jersey (1967)

    MATH  Google Scholar 

  3. Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems. Princeton University Press, Princeton (1983)

    MATH  Google Scholar 

  4. Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Clarendon Press, oxford (1993)

    MATH  Google Scholar 

  5. Kilpeläinen, T.: Hölder continuity of solutions to quasilinear elliptic equations involving measures. Potential Anal 3, 265–272 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kilpeläinen, T., Zhong, X.: Removable sets for continuous solutions of quasilinear elliptic equations. Proc. Am. Math. Soc. 130, 1681–1688 (2002)

    Article  MATH  Google Scholar 

  7. Maeda, F.-Y.: Renormalized solutions of Dirichlet problems for quasilinear elliptic equations with general measure data. Hiroshima Math. J. 38(1), 51–93 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Maeda, F.-Y., Ono, T.: Resolutivity of ideal boundary for nonlinear Dirichlet problems. J. Math. Soc. Japan 52, 561–581 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Maeda, F.-Y., Ono, T.: Properties of harmonic boundary in nonlinear potential theory. Hiroshima Math. J. 30, 513–523 (2000)

    MathSciNet  MATH  Google Scholar 

  10. Maeda, F.-Y., Ono, T.: Perturbation theory for nonlinear Dirichlet problems. Ann. Acad. Sci. Fenn. Math. 28, 207–222 (2003)

    MathSciNet  MATH  Google Scholar 

  11. Mikkonen, P.: On the Wolff potential and quasilinear elliptic equations involving measures. Ann. Acad. Sci. Fenn. Math. Dissertationes 104, 1–71 (1996)

    MathSciNet  Google Scholar 

  12. Michael, J.H., Ziemer, W.P.: Interior regularity for solutions to obstacle problems. Nonlinear Anal. Theory Method Appl. 10, 1427–1448 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ono, T.: On solutions of quasi-linear partial differential equations \(-\)div\({{{\cal A}(x,\nabla u) + {{\cal B}}}}(x,u) = 0\). RIMS Kōkyūroku 1016, 146–165 (1997)

  14. Ono, T.: Potential theory for quasi-linear partial differential equations \(-\)div\({{{\cal A}(x,\nabla u) + {{\cal B}}}}(x,u)=0\), Doctoral Thesis. Hiroshima University (2000)

  15. Ono, T.: Hölder continuity of solutions to quasilinear elliptic equations with measure data, Potential theory in Matue, 327–338, Studies in Pure Mathematics 44. Math. Soc, Japan (2006)

  16. Ono, T.: Superharmonic functions and differential equations involving measures for quasilinear elliptic operators with lower order terms. Ann. Acad. Sci. Fenn. Math. 33, 171–204 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Rakotoson, J.M., Ziemer, W.P.: Local behavior of solutions of quasilinear elliptic equations with general structure. Trans. Amer. Math. Soc. 319, 747–764 (1990)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author is grateful to Professor Fumi-Yuki Maeda for giving many valuable comments. This research has been supported by a Grant-in-Aid for Scientific Research in Japan.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Takayori Ono.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ono, T. Removable sets for Hölder continuous solutions of quasilinear elliptic equations with lower order terms. Math. Ann. 356, 355–372 (2013). https://doi.org/10.1007/s00208-012-0845-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-012-0845-6

Mathematics Subject Classification (2000)

Navigation