Skip to main content
Log in

Solutions ofLu=u α dominated byL-harmonic functions

Journal d’Analyse Mathématique Aims and scope

Abstract

LetL be a second order elliptic differential operator on a differentiable manifoldM and let 1 <α≤2. We investigate connections bewween classU of all positive solutions of the equationLu=u α and classH of all positiveL-harmonic functions (i.e., solutions of the equationsLh=0). PutuU 0 ifuU and ifuh for somehH. To everyuU 0 there corresponds the minimalL-harmonic functionh u which dominatesu andu→h u is a 1–1 mapping fromU 0 onto a subsetH 0 ofH. The inverse mapping associates with everyhH 0 the maximal element ofU dominated byh.

Supposeg(x, dy) is Green's kernel,k(x, y) is the Martin kernel and ϖM is the Martin boundary associated withL. A subset Γ of ϖM is calledR-polar if it is not hit by the rangeR of the (L, α)-superdiffusion. It is calledM-polar if\(\int\limits_M {g\left( {c,dx} \right)[\int\limits_\Gamma {k(x,y)v(dy)]^\alpha } } \) is equal to 0 or ∞ for everycM and every measure ρ.

EveryhH has a unique representation\(h(x) = \int\limits_{\partial M} {k\left( {x,y} \right)v\left( {dy} \right)} \) where ρ is a measure concentrated on the minimal partM * of ϖM.

We show that the condition:

  1. (a)

    ρ(Γ)=0 for allR sets Γ is necessary and the condition:

  2. (b)

    ρ(Γ)=0 for allM-polar sets Γ is sufficient forh to belong toH 0. IfM is a bounded domain of classC 2, λ in ℝd, then conditions (a) and (b) are equivalent and therefore each of them characterizesH 0. This was conjectured by Dynkin a few years ago and proved in a recent paper of Le Gall forL=Δ, α=2 and domains of classC 5.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. D. R. Adams and L. I. Hedberg,Function Spaces and Potential Theory, forthcoming book.

  2. P. Baras and M. Pierre,Singularités éliminable pour des équations semi-linéaires, Ann. Inst. Fourier, Grenoble34 (1984), 185–206.

    MATH  MathSciNet  Google Scholar 

  3. G. Choquet,Theory of capacities, Ann. Inst. Fourier5 (1953–54), 131–295.

    MathSciNet  Google Scholar 

  4. C. Dellacherie and P.-A. Meyer,Probabilités et potentiel, Théorie des martingales, Hermann, Paris, 1975, 1980, 1983, 1987.

    Google Scholar 

  5. N. Dunford and J. T. Schwartz,Linear Operators, Part I: General Theory, Interscience Publishers, New York, London, 1958.

    Google Scholar 

  6. E. B. Dynkin,Theory of Markov Processes, Pergamon Press, Oxford/London/New York/Paris, 1960.

    Google Scholar 

  7. E. B. Dynkin,Exit space of a Markov process [English translation: Russian Math. Surveys24, 4, pp. 89–157], Uspekhi Mat. Nauk24, 4 (148), (1969), 89–152.

    MathSciNet  Google Scholar 

  8. E. B. Dynkin,Minimal excessive measures and functions [Reprinted in: E. B. DynkinMarkov Processes and Related Problems of Analysis, London Math. Soc. Lecture Note Series 54, Cambridge University Press, Cambridge, 1982], Trans. Amer. Math. Soc.258 (1980), 217–244.

    Article  MATH  MathSciNet  Google Scholar 

  9. E. B. Dynkin,On regularity of superprocesses, Probab. Theory Rel. Fields95 (1993), 263–281.

    Article  MATH  MathSciNet  Google Scholar 

  10. E. B. Dynkin,Superprocesses and partial differential equations, Ann. Probab.21 (1993), 1185–1262.

    Article  MATH  MathSciNet  Google Scholar 

  11. E. B. Dynkin,An Introduction to Branching Measure-Valued Processes, Series, Vol. 6, American Mathematical Society, Providence, Rhode Island, 1994.

    MATH  Google Scholar 

  12. E. B. Dynkin and S. E. Kuznetsov,Superdiffusions and removable singularities for quasilinear partial differential equations, Comm. Pure Appl. Math. (to appear).

  13. P. J. Fitzsimmons,Construction and regularity of measure-valued Markov branching processes, Israel J. Math.64 (1988), 337–361.

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Gmira and L. Véron,Boundary singularities of solutions of some nonlinear elliptic equations, Duke Math. J.64 (1991), 271–324.

    Article  MATH  MathSciNet  Google Scholar 

  15. K. Itô,Stochastic differential equations in a differentiable manifold, Mem. Coll. Science, Univ. Kyoto, Ser. A28 (1953), 209–223.

    Google Scholar 

  16. J.-F. Le Gall,Les solutions positives de Δu=u 2 dans le disque unité, C. R. Acad. Sci. Paris. Sér. I317 (1993), 873–878.

    MATH  Google Scholar 

  17. J.-F. Le Gall,The Brownian snake and solutions of Δ=u 2 in a domain, Probab. Theory Rel. Fields102 (1995), 393–432.

    Article  MATH  Google Scholar 

  18. Y. C. Sheu,Removable boundary singularities for solutions of some nonlinear differential equations, Duke Math. J.74 (1994), 701–711.

    Article  MATH  MathSciNet  Google Scholar 

  19. K. Yosida,The fundamental solution of the parabolic equation in a Riemannian space, Osaka Math. J.5 (1953), 65–74.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Partially supported by National Science Foundation Grant DMS-9301315.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dynkin, E.B., Kuznetsov, S.E. Solutions ofLu=u α dominated byL-harmonic functions. J. Anal. Math. 68, 15–37 (1996). https://doi.org/10.1007/BF02790202

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02790202

Keywords

Navigation