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manuscripta mathematica

, Volume 115, Issue 2, pp 207–238 | Cite as

Green function for X-elliptic operators

  • Giovanni MazzoniEmail author
Article

Abstract.

We study the Green function associated to second order X-elliptic operators with non-regular coefficients. An existence and uniqueness theorem is given, along with some local estimations and a representation formula for the solution of the Dirichlet problem.

Keywords

Green Function Dirichlet Problem Local Estimation Uniqueness Theorem Representation Formula 
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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Notes

Acknowledgments.

It is a pleasure to thank Ermanno Lanconelli for his encouragement to my work on this problem.

References

  1. 1.
    Bony, J.-M.: Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier Grenoble 19-1, 277–304 (1969)Google Scholar
  2. 2.
    Cancelier, C., Xu, C.J.: Fonction de Green pour des opérateurs elliptiques dégénérés a coefficients non réguliers. PreprintGoogle Scholar
  3. 3.
    Cancelier, C., Xu, C.J.: Remarques sur les fonctions de Green associées aux opérateurs de Hörmander. C. R. Acad. Sci. Paris, t. 330 (Série I), 433–436 (2000)Google Scholar
  4. 4.
    Chanillo, S., Wheeden, R.L.: Existence and estimates of Green’s function for degenerate elliptic equations. Ann. Scuola Norm. Sup. Pisa 15, 309–340 (1988)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Franchi, B., Lanconelli, E.: Une métrique associée à une classe d’opérateurs elliptiques dégénérés. In: Proceedings of the meeting “Linear Partial and Pseudo Differential Operators” Torino, Rend. Sem. Mat. Univ. e Politec. Torino, 105–114 (1982)Google Scholar
  6. 6.
    Franchi, B., Lanconelli, E.: Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10 (4), 523–541 (1983)zbMATHGoogle Scholar
  7. 7.
    Franchi, B., Lanconelli, E.: An embedding theorem for Sobolev Spaces related to non-smooth vector fields and Harnack inequality. Comm. Partial Diff. Eqs. 9 (13), 1237–1264 (1984)zbMATHGoogle Scholar
  8. 8.
    Franchi, B., Serapioni, R., Serra Cassano, F.: Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields. Houston J. Math. 22 (4), 859–890 (1996)zbMATHGoogle Scholar
  9. 9.
    Franchi, B., Serapioni, R., Serra Cassano, F.: Approximation and imbedding theorems for weighted Sobolev Spaces associated with Lipschitz continuous vector fields. Bollettino U.M.I. 11-B (7), 83–117 (1997)Google Scholar
  10. 10.
    Friedrichs, K.O.: The identity of weak and strong extensions of differential operators. Trans. Am. Math. Soc. 55, 132–151 (1944)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Garofalo, N., Nhieu, D.-M.: Isoperimetric and Sobolev Inequalities for Carnot-Carathéodory Spaces and the existence of minimal surfaces. Commun. Pure Appl. Math. XLIX, 1081–1144 (1996)Google Scholar
  12. 12.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Berlin-Heidelberg-New York: Springer-Verlag, 1998Google Scholar
  13. 13.
    Gutiérrez, C.E., Lanconelli, E.: Maximum principle, non-homogeneous Harnack inequality, and Liouville theorems for X-elliptic operators. Comm. Partial Diff. Eqs. 28 (11–12), 1833–1862 (2003)Google Scholar
  14. 14.
    Grüter, C.E., Widman, K.O.: The Green function for uniformly elliptic equations. Manuscripta Math. 37, 303–342 (1982)MathSciNetGoogle Scholar
  15. 15.
    John F., Nirenberg, L.: On functions of Bounded Mean Oscillation. Commun. Pure Appl. Math. XIV, 415–426 (1961)Google Scholar
  16. 16.
    Lanconelli, E.: Alcuni aspetti dell’Analisi Reale negli spazi omogenei. Regolarizzazione delle soluzioni deboli delle equazioni P-ellittiche. Seminario di Analisi Matematica A.A. 1995/96, Dip. Mat. Bologna 53–71 (1996)Google Scholar
  17. 17.
    Lanconelli, E., Morbidelli, D.: On the Poincarés ineqality for vector fields. Ark. Mat. 38, 327–342 (2000)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Littman, W., Stampacchia, G., Weinberger, H.F.: Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa (III) 17, 43–77 (1963)zbMATHGoogle Scholar
  19. 19.
    Moser, J.: A new proof of de Giorgi’s Theorem concerning the regularity problem for Elliptic Differential Equations. Commun. Pure Appl. Math. XIII, 457–468 (1960)Google Scholar
  20. 20.
    Moser, J.: On Harnack’s Theorem for Elliptic Differential Equations. Commun. Pure Appl. Math. XIV, 577–591 (1961)Google Scholar
  21. 21.
    Montanari, A., Morbidelli, D.: Balls defined by nonsmooth vector fields and the Poincarés inequality. Preprint (2003)Google Scholar
  22. 22.
    Salinas, O.: Harnack inequality and Green function for a certain class of degenerate elliptic differential operators. Revista Matemática Iberoamericana 7 (3), 313–349 (1991)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di BolognaBolognaItalia

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