Skip to main content
SpringerLink
Log in

Semismall perturbations in the martin theory for elliptic equations

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We investigate stability of Martin boundaries for positive solutions of elliptic partial differential equations. We define a perturbation which isG DL -semismall at infinity, show that Martin boundaries are stable under this perturbation, and give sufficient conditions for it.

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.

Similar content being viewed by others

References

  1. S. Agmon,On positive solutions of elliptic equations with periodic coefficients in ℝn,spectral results and extensions to elliptic operators on Riemannian manifolds, Proceedings of an International Conference on Differential Equations (I. W. Knowles and R. T. Lewis, eds.), North-Holland, Amsterdam, 1984, pp. 7–17.

    Google Scholar 

  2. H. Aikawa,On the upper bounds of Green potentials, Hiroshima Mathematical Journal24 (1994), 607–612.

    MATH  MathSciNet  Google Scholar 

  3. H. Aikawa and M. Murata,Generalized Cranston-McConnell inequalities and Martin boundaries of unbounded domains, Journal d’Analyse Mathématique69 (1996), 137–152.

    MATH  MathSciNet  Google Scholar 

  4. D. G. Aronson,Non-negative solutions of linear parabolic equations, Annali della Scuola Normale Superiore di Pisa22 (1968), 607–694.

    MATH  MathSciNet  Google Scholar 

  5. R. Bañuelos,On an estimate of Cranston and McConnell for elliptic diffusions in uniform domains, Probability Theory and Related Fields76 (1987), 311–323.

    Article  MATH  MathSciNet  Google Scholar 

  6. R. Bañuelos,Intrinsic ultracontractivity and eigenfunction estimates for Schrödinger operators, Journal of Functional Analysis100 (1991), 181–206.

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Bañuelos,Lifetime and heat kernel estimates in non-smooth domains, inPartial Differential Equations with Minimal Smoothness and Applications (B. Dahlberg et al., eds.), Springer-Verlag, New York, 1992, pp. 37–48.

    Google Scholar 

  8. R. Bañuelos and T. Carroll,Conditioned Brownian motion and hyperbolic geodesics in simply connected domains, The Michigan Mathematical Journal40 (1993), 321–332.

    Article  MATH  MathSciNet  Google Scholar 

  9. R. Bañuelos and B. Davis,Heat kernel, eigenfunctions, and conditioned Brownian motion in planar domains, Journal of Functional Analysis84 (1989), 188–200.

    Article  MATH  MathSciNet  Google Scholar 

  10. R. Bañuelos and B. Davis,A geometric characterization of intrinsic ultracontractivity for planar domains with boundaries given by the graphs of functions, Preprint, Purdue University, December 1991.

  11. A. Bouchricha, W. Hansen and H. Hueber,Continuous solutions of the generalized Schrödinger equation and perturbation of harmonic spaces, Expositiones Mathematicae5 (1987), 97–135.

    MathSciNet  Google Scholar 

  12. R. Bañuelos and B. Øksendal,Exit times for elliptic diffusions and BMO, Proceedings of the Edinburgh Mathematical Society30 (1987), 273–287.

    Article  MATH  MathSciNet  Google Scholar 

  13. F. Ciapriani,Intrinsic ultracontractivity of Dirichlet Laplacians in non-smooth domains, Potential Analysis3 (1994), 203–218.

    Article  MathSciNet  Google Scholar 

  14. C. Constantinescu and A. Cornea,Potential Theory on Harmonic Spaces, Springer-Verlag, Berlin, 1972.

    MATH  Google Scholar 

  15. M. Cranston, E. Fabes and Z. Zhao,Conditional gauge and potential theory for the Schrödinger operator, Transactions of the American Mathematical Society307 (1988), 171–194.

    Article  MATH  MathSciNet  Google Scholar 

  16. M. Cranston and T. R. McConnell,The life time of conditioned Brownian motion, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete65 (1983), 1–11.

    Article  MATH  MathSciNet  Google Scholar 

  17. E. B. Davies,Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989.

    MATH  Google Scholar 

  18. E. B. Davies and B. Simon,Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians, Journal of Functional Analysis59 (1984), 335–395.

    Article  MATH  MathSciNet  Google Scholar 

  19. N. Falkner,Conditional Brownian motion in rapidly exhaustible domains, The Annals of Probability15 (1987), 1501–1514.

    Article  MATH  MathSciNet  Google Scholar 

  20. M. Grüter and K-O. Widman,The Green function for uniformly elliptic equations, Manuscripta Mathematica37 (1982), 303–342.

    Article  MATH  MathSciNet  Google Scholar 

  21. L. L. Helms,Introduction to Potential Theory, Wiley-Interscience, New York, 1969.

    MATH  Google Scholar 

  22. I. W. Herbst and Z. Zhao,Green functions for the Schrödinger equation with short range potentials, Duke Mathematical Journal59 (1989), 475–519.

    Article  MATH  MathSciNet  Google Scholar 

  23. K. Kurata,Continuity and Harnack’s inequality for solutions of elliptic partial differential equations of second order, Indiana University Mathematics Journal43 (1994), 411–440.

    Article  MATH  MathSciNet  Google Scholar 

  24. P. A. Loeb,An axiomatic treatment of pairs of elliptic differential equations, Annales de l’Institut Fourier16 (1966), 167–208.

    MATH  MathSciNet  Google Scholar 

  25. V. Lin and Y. Pinchover,Manifolds with group actions and elliptic operators, Memoirs of the American Mathematical Society112 (1994), No. 540.

    Google Scholar 

  26. W. Littman, G. Stampacchia, and H. F. Weinberger,Regular points for elliptic equations with discontinuous coefficients, Annali della Scuola Normale Superiore di Pisa17 (1963), 43–77.

    MATH  MathSciNet  Google Scholar 

  27. F-Y. Maeda,Harmonic and full-harmonic structures on a differential manifold, Journal of Science of Hiroshima University, Series A-I34 (1970), 271–312.

    MATH  Google Scholar 

  28. R. S. Martin,Minimal positive harmonic functions, Transactions of the American Mathematical Society49 (1941), 137–172.

    Article  MATH  MathSciNet  Google Scholar 

  29. M. Murata,Structure of positive solutions to (−Δ + V)u=0in ℝn, Duke Mathematical Journal53 (1986), 869–943.

    Article  MATH  MathSciNet  Google Scholar 

  30. M. Murata,On construction of Martin boundaries for second order elliptic equations, Publications of the Research Institute for Mathematical Sciences of Kyoto University26 (1990), 585–627.

    Article  Google Scholar 

  31. M. Murata,Martin compactification and asymptotics of Green functions for Schrödinger operators with anisotropic potentials, Mathematische Annalen288 (1990), 211–230.

    Article  MATH  MathSciNet  Google Scholar 

  32. M. Murata,Uniform restricted parabolic Harnack inequality, separation principle, and ultracontractivity for parabolic equations, inFunctional Analysis and Related Topics, 1991, Lecture Notes in Mathematics, Vol. 1540 (H. Komatsu, ed.), Springer-Verlag, Berlin, 1993, pp. 277–288.

    Chapter  Google Scholar 

  33. M. Murata,Non-uniqueness of the positive Cauchy problem for parabolic equations, Journal of Differential Equations123 (1995), 343–387.

    Article  MATH  MathSciNet  Google Scholar 

  34. M. Murata,Non-uniqueness of the positive Dirichlet problem for parabolic equations in cylinders, Journal of Functional Analysis135 (1996), 456–487.

    Article  MATH  MathSciNet  Google Scholar 

  35. M. Nakai,Comparison of Martin boundaries for Schrödinger operator, Hokkaido Mathematical Journal18 (1989), 245–261.

    MATH  MathSciNet  Google Scholar 

  36. Y. Pinchover,On positive solutions of second order elliptic equations, stability results and classification, Duke Mathematical Journal57 (1988), 955–980.

    Article  MATH  MathSciNet  Google Scholar 

  37. Y. Pinchover,Criticality and ground states for second order elliptic equations, Journal of Differential Equations80 (1989), 237–250.

    Article  MATH  MathSciNet  Google Scholar 

  38. Y. Pinchover,On the localizations of binding for Schrödinger operators and its extension to elliptic operators, Journal d’Analyse Mathématique66 (1995), 57–87.

    MATH  MathSciNet  Google Scholar 

  39. R. G. Pinsky,Positive harmonic functions and diffusions, Cambridge University Press, Cambridge, 1995.

    Google Scholar 

  40. G. Stampacchia,Le problème de Dirichlet pour les équations elliptique de second ordre à coefficients discontinuous, Annales de l’Institut Fourier15 (1965), 189–257.

    MathSciNet  MATH  Google Scholar 

  41. J. C. Taylor,The Martin boundaries of equivalent sheaves, Annales de l’Institut Fourier20 (1970), 433–456.

    MATH  Google Scholar 

  42. Z. Zhao,Green function for Schrödinger operator and conditioned Feynman-Kac gauge, Journal of Mathematical Analysis and Applications116 (1986), 309–334.

    Article  MATH  MathSciNet  Google Scholar 

  43. Z. Zhao,Subcriticality and gaugeability of the Schrödinger operator, Transactions of the American Mathematical Society334 (1992), 75–96.

    Article  MATH  MathSciNet  Google Scholar 

  44. Z. Zhao,On the existence of positive solutions of nonlinear elliptic equations — A probabilistic potential theory approach, Duke Mathematical Journal69 (1994), 247–258.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, 152, Tokyo, Japan

    Minoru Murata

Authors
  1. Minoru Murata
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Minoru Murata.

Additional information

This work was carried out partially while the author was visiting the Department of Mathematics, Technion — Israel Institute of Technology. He wishes to thank the Israel Association for the Promotion of International Scientific Relations, the Japan Society of Promotion of Science, and the Technion.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Murata, M. Semismall perturbations in the martin theory for elliptic equations. Isr. J. Math. 102, 29–60 (1997). https://doi.org/10.1007/BF02773794

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02773794

Keywords

Search

Navigation