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Potential Analysis

, Volume 13, Issue 1, pp 81–102 | Cite as

Inequalities for Green Functions in a Dini-Jordan Domain in R2

  • Mohamed Selmi
Article

Abstract

We establish inequalities for Green functions of Dini-smooth Jordan domains in R2.We give a version of the 3G theorem for these domains. With the help of these results, we prove comparison theorems between the Green kernel of Δ and the Green kernel of Δ − μ where μ is a nonnegative Radon measure.

Green function Jordan domain comparison perturbed Green function 

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References

  1. 1.
    Ahlfors, L. V.: Complex Analysis, McGraw-Hill Book Company (2nd edn), 1966.Google Scholar
  2. 2.
    Ancona, A.: ‘Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien’, Ann. Inst. Fourier 28(4) (1978), 169–213.Google Scholar
  3. 3.
    Ancona, A.: ‘Comparaison des mesures harmoniques et des fonctions de Green pour des opérateurs elliptiques sur un domain lipschitzien’, CRAS 294 (1982), 505–508.Google Scholar
  4. 4.
    Ben Saad, H.: Généralisation des Noyaux V h et Applications, Séminaire de théorie du potentiel de Paris N° 7. Lecture Notes in Math. N° 1061, Springer-Verlag, 1984.Google Scholar
  5. 5.
    Boboc, N. and Bucur, G. H.: Perturbations in Excessive Structures Complex Analysis, Lecture Notes in Math. 1014, Fifth Romanian Finnish Seminar, Part 2, Bucharest, 1981.Google Scholar
  6. 6.
    Chung, K. L.: Probability Method in Potential Theory, Lecture Notes in Math. 1344, Springer, Berlin, 1987, pp. 42–54.Google Scholar
  7. 7.
    Chung, K. L. and Zhao, Z.: From Brownian Motion to Schrödinger's Equation, Springer-Verlag, New York, Berlin, Heidelberg, 1995.Google Scholar
  8. 8.
    Curtiss, J. H.: Introduction to Functions of Complex Variable, M. Dekker, New York, 1978.Google Scholar
  9. 9.
    Hirsch, F.: ‘Conditions nécessaires et suffisantes d'existence de résolvantes’, Z. Wahrsch. Verw. Gebiete 29 (1974), 73–85.Google Scholar
  10. 10.
    Hueber, H. and Sieveking, M.: ‘Uniform bounds for quotient of Green functions on C 1;1 domains’, Ann. Inst. Fourier 32(1) (1982), 105–117.Google Scholar
  11. 11.
    Maagli, H. and Selmi, M.: ‘Perturbation et comparaison des semi-groupes’, Revue Roum. de Math. Pures et Appliquées 34(1) (1989), 29–40.Google Scholar
  12. 12.
    Maagli, H. and Selmi, M.: ‘Perturbation des résolvantes et des semi-groupes par une mesure de Radon positive’, Math. Zeitschrift 205 (1990), 379–393.Google Scholar
  13. 13.
    McConnell, T.R.: ‘A conformal inequality related to the conditional gauge theorem’, Trans. Am. Math. Soc. 318(2) (1990), 721–733.Google Scholar
  14. 14.
    Pommerenke, Ch.: Boundary Behavior of Conformal Maps, Springer-Verlag, 1991.Google Scholar
  15. 15.
    Ransford, T.: Potential Theory in the Complex Plane, Cambridge University Press, London, 1995.Google Scholar
  16. 16.
    Selmi, M.: Critère de comparaison de Certains Noyaux de Green, Séminaire de Théorie du Potentiel de Paris N° 8, Lecture Notes in Math. 1235, 1987.Google Scholar
  17. 17.
    Selmi, M.: ‘Comparaison des noyaux de Green sur les domaines C 1;1’, Revue Roum. de Math. Pures et Appliquées 36(1–2) (1991), 91–100.Google Scholar
  18. 18.
    Selmi, M.: ‘Critères de comparaison des noyaux de Green sur certains domains de ℝ2’. Revue Roum. de Math. Pures et Appliquées 42(3–4) (1997), 320–337.Google Scholar
  19. 19.
    Zhao, Z.: Green Functions and Conditioned Gauge Theorem for a 2-Dimensional Domains, Seminar on stochastic processes, Birkhaüser, Boston, 1988.Google Scholar
  20. 20.
    Zhao, Z.: Gaugeability for Unbounded Domain, Seminar on stochastic processes, Birkhaüser, Boston, 1989.Google Scholar
  21. 21.
    Zhao, Z.: ‘An equivalence theorem for Schrödinger operators and its applications’. In Diffusion Processes and Related Problems in Analysis, Vol. 1, Birkhaüser, Boston, 1990.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Mohamed Selmi
    • 1
  1. 1.Faculty of Sciences of Tunis, Department of Mathematics, Campus UniversitaireUniversity of Tunis-IITunisTunisia

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