Skip to main content
SpringerLink
Log in

Harnack inequality for a class of strongly degenerate elliptic operators formed by Hörmander vector fields

  • Published:
Manuscripta Mathematica Aims and scope Submit manuscript

Abstract

We prove Harnack inequality and regularity for solutions of a linear strongly degenerate elliptic equation. We assume the lower order terms in Stummel–Kato classes with respect to the Carnot–Carathéodory metric. We follow the pattern by Serrin in Acta Math 111:247–302, (1964).

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aizenman M., Simon B.: Brownian motion and Harnack’s inequality for Schrdinger operators. Commun. Pure Appl. Math. 35, 209–271 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  2. Buckley S.M.: Inequalities of John-Nirenberg type in doubling spaces. J. Anal. Math. 79, 215–240 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chiarenza F., Fabes E., Garofalo N.: Harnack’s inequality for Schrödinger operators and the continuity of solutions. Proc. Am. Math. Soc. 98(3), 415–425 (1986)

    MATH  MathSciNet  Google Scholar 

  4. Citti G., Di Fazio G.: Hölder continuity of the solutions for operators which are a sum of squares of vector fields plus a potential. Proc. Am. Math. Soc. 122, 741–750 (1994)

    MATH  MathSciNet  Google Scholar 

  5. Citti G., Garofalo N., Lanconelli E.: Harnack’s inequality for sum of squares of vector fields plus a potential. Am. J. Math. 115, 699–734 (1994)

    Article  MathSciNet  Google Scholar 

  6. Di Fazio G.: Hölder-continuity of solutions for some Schrödinger equations. Rend. Semin. Mat. Univ. Padova 79, 173–183 (1988)

    MATH  MathSciNet  Google Scholar 

  7. Di Fazio G.: Poisson equation in Morrey spaces. J. Math. Anal. Appl. 163, 157–167 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  8. Di Fazio G.: Dirichlet problem characterization of regularity. Manuscr. Math. 84, 47–56 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  9. Di Fazio G., Zamboni P.: Hölder continuity for quasilinear subelliptic equations in Carnot Carath认ory spaces. Math. Nachr. 272, 3–10 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Di Fazio G., Zamboni P.: Local regularity of solutions to quasilinear subelliptic equations in Carnot Carathéodory spaces. Boll. Unione Mat. Ital. (8) 9B(2), 485–504 (2006)

    MathSciNet  Google Scholar 

  11. Di Fazio G., Fanciullo M.S., Zamboni P.: Harnack inequality and smoothness for quasilinear degenerate elliptic equations. J. Differ. Equ. 245(10), 2939–2957 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fabes E., Kenig C., Serapioni R.: The local regularity of solutions of degenerate elliptic equations. Commun. Partial Differ. Equ. 7(1), 77–116 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gutierrez C.E.: Harnack’s inequality for degenerate Schrödinger operators. Trans. AMS 312, 403–419 (1989)

    Article  MATH  Google Scholar 

  14. Kurata K.: Continuity and Harnack’s inequality for solutions of elliptic partial differential equations of second order. Indiana Univ. Math. J. 43(2), 411–440 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lu G.: Weighted Poincaré and Sobolev inequality for vector fields satisfying Hörmander’s condition and its application. Rev. Iberoam. 8(3), 367–439 (1992)

    MATH  Google Scholar 

  16. Lu G.: On Harnack’s inequality for a class of strongly degenerate Schrödinger operators formed by vector fields. Differ. Integral Equ. 7(1), 73–100 (1994)

    MATH  Google Scholar 

  17. Mohammed A.: Hölder continuity of solutions of some degenerate elliptic differential equations. Bull. Austral. Math. Soc. 62, 369–377 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  18. Nagel A., Stein E.M., Wainger S.: Balls and metrics defined by vector fields. I. Basic properties. Acta Math. 155(1–2), 103–147 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  19. Rakotoson J.M., Ziemer W.P.: Local behavior of solutions of quasilinear elliptic equations with general structure. Trans. Am. Math. Soc. 319(2), 747–764 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  20. Serrin J.: Local behaviour of solutions of quasilinear equations. Acta Math. 111, 247–302 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  21. Simader C.G.: An elementary proof of Harnack’s inequality for Schrödinger operators and related topics. Math. Z. 203(1), 129–152 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  22. Vitanza C., Zamboni P.: Necessary and sufficient conditions for Hölder continuity of solutions of degenerate Schrdinger operators. Le Mat. 52(2), 393–409 (1997)

    MATH  MathSciNet  Google Scholar 

  23. Zamboni P.: Local boundedness of solutions of quasilinear elliptic equations with coefficients in Morrey spaces. Boll. Unione Mat. Ital. (7) 8B(4), 985–997 (1994)

    MathSciNet  Google Scholar 

  24. Zamboni P.: Local behavior of solutions of quasilinear elliptic equations with coefficients in Morrey spaces. Rend. Mat. Appl. (7) 15(2), 251–262 (1995)

    MATH  MathSciNet  Google Scholar 

  25. Zamboni P.: The Harnack inequality for quasilinear elliptic equations under minimal assumptions. Manuscr. Math. 102(3), 311–323 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  26. Zamboni P.: Hölder continuity for solutions of linear degenerate elliptic equations under minimal assumptions. J. Differ. Equ. 182(1), 121–140 (2002)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125, Catania, Italy

    G. Di Fazio, M. S. Fanciullo & P. Zamboni

Authors
  1. G. Di Fazio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. S. Fanciullo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. Zamboni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Di Fazio.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Di Fazio, G., Fanciullo, M.S. & Zamboni, P. Harnack inequality for a class of strongly degenerate elliptic operators formed by Hörmander vector fields. manuscripta math. 135, 361–380 (2011). https://doi.org/10.1007/s00229-010-0420-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-010-0420-y

Mathematics Subject Classification (2000)

Search

Navigation