Skip to main content
SpringerLink
Log in

Some Aspects of the Boundary Trace Problem for Solutions of Nonlinear Elliptic Equations

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

The boundary trace problem for positive solutions of -Δu + g(x, u) = 0 is considered for a large class of nonlinearities and three different methods for defining the trace are compared. The boundary trace is usually a generalized Borel measure. The associated Dirichlet problem with boundary data in the set of such Borel measures is studied.

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.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. C.Bandle and M.Marcus,"Asymptotic behavior of solutions and their derivative for semilinear elliptic problems with blow-up on the boundary,"Ann.Inst.Henri Poincaré,12 155-171 (1995).

    Google Scholar 

  2. C.Bandle and M.Marcus,"Large solutions of semilinear elliptic equations:existence,uniqueness and asymptotic behavior,"J.Anal.Math.,58 9-24 (1992).

    Google Scholar 

  3. P.Baras and M.Pierre,"Singularitéséliminables pour deséquations semi-linéaires,"Ann.Inst. Fourier,34 185-206 (1984).

    Google Scholar 

  4. P.Baras and J.Goldstein,"The heat equation with a singular potential,"Trans.Am.Math.Soc., 284 121-139 (1984).

    Google Scholar 

  5. P.Benilan and H.Brezis,Nonlinear problems related to the Thomas-Fermi equation,Unpublished paper,see [8 ].

  6. M.F. Bidaut-Véron and L.Vivier,"An elliptic semilinear equation with source term involving bound-ary measures:the subcritical case,"Rev.Mat.Iberoam.,16 477-513 (2000).

    Google Scholar 

  7. H.Brezis,Uneéquation semi-linéaire avec conditions aux limites dans L 1,Unpublished paper,see also [38,Chap.4 ].

  8. H.Brezis,"Some variational problems of the Thomas-Fermi type,"In:R.W. Cottle, F.Gianessi, and J.-L.Lions (Eds.),Variational Inequalities,Wiley, Chichester (1980),pp.53-73.

  9. X.Cabre,Extremal solutions and instantaneous complete blow-up for elliptic and parabolic problems, To appear.

  10. R.Dautray and J.-L.Lions,Analyse Mathématique et Calcul Numérique,Masson,Paris (1987).

  11. J.Doob,Classical Potential Theory and Its Probabilistic Counterpart,Springer-Verlag, Berlin-New-York (1984).

  12. Y.Du and Z.Guo,The degenerate logistic model and a singularly mixed boundary blow-up problem, To appear.

  13. E.B. Dynkin and S.E. Kuznetsov,"Trace on the boundary for solutions of nonlinear differential equations,"Trans.Am.Math.Soc.,350 4499-4519 (1998).

    Google Scholar 

  14. E.B. Dynkin and S.E. Kuznetsov,"Solutions of nonlinear differential equtions on a Riemannian manifold and their trace on the Martin boundary,'Trans.Am.Math.Soc.,350 4521-4552 (1998).

    Google Scholar 

  15. E.B. Dynkin and S.E. Kuznetsov,"Fine topology and ne trace on the boundary associated with a class of quasilinear differential equations,"Commun.Pure Appl.Math.,5 No.1,897-936 (1998).

    Google Scholar 

  16. J.Fabbri and J.R. Licois,"Behavior at boundary of solutions of a weakly superlinear elliptic equa-tion,"Adv.Nonlinear Studies,2 147-176 (2002).

    Google Scholar 

  17. D.Gilbarg and N.S. Trudinger,Partial Differential Equations of Second Order,2nd ed.,Springer-Verlag, Berlin-New York (1983).

    Google Scholar 

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

    Google Scholar 

  19. M.Grillot and L. Véron,"Boundary trace of solutions of the prescribed Gaussian curvature equation,"Proc.R. Soc.Edinburgh,Sect.A,Math.,130 A 1-34 (2000).

    Google Scholar 

  20. I.Iscoe,"On the support of measure-valued critical branching Brownian motion,"Ann.Probab.,16 200-221 (1988).

    Google Scholar 

  21. J.B. Keller,"On solutions of u =f (u ),"Commun.Pure Appl.Math.,10 503-510 (1957).

  22. J.F. Le Gall,"Les solutions positives de u =u 2 dans le disque unité,"C.R. Acad.Sci.,Paris, Sér.I,Math.,317 873-878 (1993).

    Google Scholar 

  23. J.F. Le Gall,"The brownian snake and solutions of u =u 2 in a domain,"Probab.Theory Relat. Fields,102 393-432 (1995).

    Google Scholar 

  24. M.Marcus and L.Véron,"Uniqueness and asymptotic behaviour of solutions with boundary blow-up for a class of nonlinear elliptic equations,"Ann.Inst.Henri Poincaré,14 237-274 (1997).

    Google Scholar 

  25. M.Marcus and L.Véron,"Traces au bord des solutions positives d 'équations elliptiques non-linéaires,"C.R. Acad.Sci.,Paris,Sér.I,Math.,321 179-184 (1995).

    Google Scholar 

  26. M.Marcus and L.Véron,"Traces au bord des solutions positives d 'équations elliptiques et paraboliques non-linéaires:résultats d 'existence et d 'unicité,"C.R. Acad.Sci.,Paris,Sér.I,Math., 323 603-608 (1996).

    Google Scholar 

  27. M.Marcus and L.Véron,"The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case,"Arch.Ration.Mech.Anal.,144 201-231 (1998).

    Google Scholar 

  28. M.Marcus and L.Véron,"The boundary trace of positive solutions of semilinear elliptic equations: the supercritical case,"J.Math.Pures Appl.,IX.S`er.77 481-524 (1998).

    Google Scholar 

  29. M.Marcus and L.Véron,"Removable singularities and boundary traces,"J.Math.Pures Appl., IX.Sér.,80 879-900 (2001).

    Google Scholar 

  30. M.Marcus and L.Véron,"The boundary trace and generalized B.V. P.for semilinear elliptic equations with a strong absorption,"Commun.Pure Appl.Math.,To appear.

  31. M.Marcus and L. Véron,Boundary trace of positive solutions of nonlinear elliptic inequalities, To appear.

  32. M.Marcus and L.Véron,"Initial trace of positive solutions to semilinear parabolic inequalities,"Adv.Nonlinear Studies,2 395-436 (2002).

    Google Scholar 

  33. B.Mselati,Classi cation et représentation probabilistes des solutions positives de u =u 2 dans un domaine, Ph.D.Thesis,Université Paris 6 (2002).

  34. R.Osserman,"On the inequality u ¡Ýf (u ),"Pac.J. Math.,7 1641-1647 (1957).

    Google Scholar 

  35. A.Ratto, M.Rigoli,and L. Véron,"Scalar curvature and conformal deformation of hyperbolic space,"J.Funct.Anal.,121 15-77 (1994).

    Google Scholar 

  36. Y.Richard and L.Véron,"Isotropic singularities of nonlinear elliptic inequalities,"Ann.Inst.Henri Poincaré,6 37-72 (1989).

    Google Scholar 

  37. J.L. Vazquez,"An a priori interior estimate for the solution of a nonlinear problem representing weak diffusion,"Nonlinear Anal.,Theory Methods Appl.,5 119-135 (1981).

  38. L.Véron,"Singularities of solutions of second order quasilinear equations,"Pitman Research Notes in Math.,353 Addison-Wesley-Longman (1996).

  39. L.Véron,"Semilinear elliptic equations with uniform blow-up on the boundary,"J.Anal.Math.,59 231-250 (1992).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 6083, Université Fransois Rabelais, Tours, 37200, France

    Laurent Véron

Authors
  1. Laurent Véron
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Véron, L. Some Aspects of the Boundary Trace Problem for Solutions of Nonlinear Elliptic Equations. Journal of Mathematical Sciences 124, 5163–5175 (2004). https://doi.org/10.1023/B:JOTH.0000047251.59557.ca

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:JOTH.0000047251.59557.ca

Keywords

Search

Navigation