Israel Journal of Mathematics

, Volume 102, Issue 1, pp 29–60

Semismall perturbations in the martin theory for elliptic equations

Article

Abstract

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

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References

  1. [Ag]
    S. Agmon,On positive solutions of elliptic equations with periodic coefficients inn,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. [Ai]
    H. Aikawa,On the upper bounds of Green potentials, Hiroshima Mathematical Journal24 (1994), 607–612.MATHMathSciNetGoogle Scholar
  3. [AM]
    H. Aikawa and M. Murata,Generalized Cranston-McConnell inequalities and Martin boundaries of unbounded domains, Journal d’Analyse Mathématique69 (1996), 137–152.MATHMathSciNetGoogle Scholar
  4. [AR]
    D. G. Aronson,Non-negative solutions of linear parabolic equations, Annali della Scuola Normale Superiore di Pisa22 (1968), 607–694.MATHMathSciNetGoogle Scholar
  5. [B1]
    R. Bañuelos,On an estimate of Cranston and McConnell for elliptic diffusions in uniform domains, Probability Theory and Related Fields76 (1987), 311–323.MATHCrossRefMathSciNetGoogle Scholar
  6. [B2]
    R. Bañuelos,Intrinsic ultracontractivity and eigenfunction estimates for Schrödinger operators, Journal of Functional Analysis100 (1991), 181–206.MATHCrossRefMathSciNetGoogle Scholar
  7. [B3]
    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. [BC]
    R. Bañuelos and T. Carroll,Conditioned Brownian motion and hyperbolic geodesics in simply connected domains, The Michigan Mathematical Journal40 (1993), 321–332.MATHCrossRefMathSciNetGoogle Scholar
  9. [BD1]
    R. Bañuelos and B. Davis,Heat kernel, eigenfunctions, and conditioned Brownian motion in planar domains, Journal of Functional Analysis84 (1989), 188–200.MATHCrossRefMathSciNetGoogle Scholar
  10. [BD2]
    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.Google Scholar
  11. [BHH]
    A. Bouchricha, W. Hansen and H. Hueber,Continuous solutions of the generalized Schrödinger equation and perturbation of harmonic spaces, Expositiones Mathematicae5 (1987), 97–135.MathSciNetGoogle Scholar
  12. [BØ]
    R. Bañuelos and B. Øksendal,Exit times for elliptic diffusions and BMO, Proceedings of the Edinburgh Mathematical Society30 (1987), 273–287.MATHMathSciNetCrossRefGoogle Scholar
  13. [Ci]
    F. Ciapriani,Intrinsic ultracontractivity of Dirichlet Laplacians in non-smooth domains, Potential Analysis3 (1994), 203–218.CrossRefMathSciNetGoogle Scholar
  14. [CC]
    C. Constantinescu and A. Cornea,Potential Theory on Harmonic Spaces, Springer-Verlag, Berlin, 1972.MATHGoogle Scholar
  15. [CFZ]
    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.MATHCrossRefMathSciNetGoogle Scholar
  16. [CM]
    M. Cranston and T. R. McConnell,The life time of conditioned Brownian motion, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete65 (1983), 1–11.MATHCrossRefMathSciNetGoogle Scholar
  17. [D]
    E. B. Davies,Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989.MATHGoogle Scholar
  18. [DS]
    E. B. Davies and B. Simon,Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians, Journal of Functional Analysis59 (1984), 335–395.MATHCrossRefMathSciNetGoogle Scholar
  19. [F]
    N. Falkner,Conditional Brownian motion in rapidly exhaustible domains, The Annals of Probability15 (1987), 1501–1514.MATHCrossRefMathSciNetGoogle Scholar
  20. [GW]
    M. Grüter and K-O. Widman,The Green function for uniformly elliptic equations, Manuscripta Mathematica37 (1982), 303–342.MATHCrossRefMathSciNetGoogle Scholar
  21. [H]
    L. L. Helms,Introduction to Potential Theory, Wiley-Interscience, New York, 1969.MATHGoogle Scholar
  22. [HZ]
    I. W. Herbst and Z. Zhao,Green functions for the Schrödinger equation with short range potentials, Duke Mathematical Journal59 (1989), 475–519.MATHCrossRefMathSciNetGoogle Scholar
  23. [K]
    K. Kurata,Continuity and Harnack’s inequality for solutions of elliptic partial differential equations of second order, Indiana University Mathematics Journal43 (1994), 411–440.MATHCrossRefMathSciNetGoogle Scholar
  24. [L]
    P. A. Loeb,An axiomatic treatment of pairs of elliptic differential equations, Annales de l’Institut Fourier16 (1966), 167–208.MATHMathSciNetGoogle Scholar
  25. [LP]
    V. Lin and Y. Pinchover,Manifolds with group actions and elliptic operators, Memoirs of the American Mathematical Society112 (1994), No. 540.Google Scholar
  26. [LSW]
    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.MATHMathSciNetGoogle Scholar
  27. [Mae]
    F-Y. Maeda,Harmonic and full-harmonic structures on a differential manifold, Journal of Science of Hiroshima University, Series A-I34 (1970), 271–312.MATHGoogle Scholar
  28. [Mar]
    R. S. Martin,Minimal positive harmonic functions, Transactions of the American Mathematical Society49 (1941), 137–172.MATHCrossRefMathSciNetGoogle Scholar
  29. [M1]
    M. Murata,Structure of positive solutions to (−Δ + V)u=0inn, Duke Mathematical Journal53 (1986), 869–943.MATHCrossRefMathSciNetGoogle Scholar
  30. [M2]
    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.CrossRefGoogle Scholar
  31. [M3]
    M. Murata,Martin compactification and asymptotics of Green functions for Schrödinger operators with anisotropic potentials, Mathematische Annalen288 (1990), 211–230.MATHCrossRefMathSciNetGoogle Scholar
  32. [M4]
    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.CrossRefGoogle Scholar
  33. [M5]
    M. Murata,Non-uniqueness of the positive Cauchy problem for parabolic equations, Journal of Differential Equations123 (1995), 343–387.MATHCrossRefMathSciNetGoogle Scholar
  34. [M6]
    M. Murata,Non-uniqueness of the positive Dirichlet problem for parabolic equations in cylinders, Journal of Functional Analysis135 (1996), 456–487.MATHCrossRefMathSciNetGoogle Scholar
  35. [N]
    M. Nakai,Comparison of Martin boundaries for Schrödinger operator, Hokkaido Mathematical Journal18 (1989), 245–261.MATHMathSciNetGoogle Scholar
  36. [P1]
    Y. Pinchover,On positive solutions of second order elliptic equations, stability results and classification, Duke Mathematical Journal57 (1988), 955–980.MATHCrossRefMathSciNetGoogle Scholar
  37. [P2]
    Y. Pinchover,Criticality and ground states for second order elliptic equations, Journal of Differential Equations80 (1989), 237–250.MATHCrossRefMathSciNetGoogle Scholar
  38. [P3]
    Y. Pinchover,On the localizations of binding for Schrödinger operators and its extension to elliptic operators, Journal d’Analyse Mathématique66 (1995), 57–87.MATHMathSciNetGoogle Scholar
  39. [Pi]
    R. G. Pinsky,Positive harmonic functions and diffusions, Cambridge University Press, Cambridge, 1995.Google Scholar
  40. [S]
    G. Stampacchia,Le problème de Dirichlet pour les équations elliptique de second ordre à coefficients discontinuous, Annales de l’Institut Fourier15 (1965), 189–257.MathSciNetMATHGoogle Scholar
  41. [T]
    J. C. Taylor,The Martin boundaries of equivalent sheaves, Annales de l’Institut Fourier20 (1970), 433–456.MATHGoogle Scholar
  42. [Z1]
    Z. Zhao,Green function for Schrödinger operator and conditioned Feynman-Kac gauge, Journal of Mathematical Analysis and Applications116 (1986), 309–334.MATHCrossRefMathSciNetGoogle Scholar
  43. [Z2]
    Z. Zhao,Subcriticality and gaugeability of the Schrödinger operator, Transactions of the American Mathematical Society334 (1992), 75–96.MATHCrossRefMathSciNetGoogle Scholar
  44. [Z3]
    Z. Zhao,On the existence of positive solutions of nonlinear elliptic equations — A probabilistic potential theory approach, Duke Mathematical Journal69 (1994), 247–258.CrossRefGoogle Scholar

Copyright information

© The Magnes Press · The Hebrew University · Jerusalem 1997

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceTokyo Institute of TechnologyTokyoJapan

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