Skip to main content

Advertisement

SpringerLink
Log in

Harnack inequality for singular fully nonlinear operators and some existence results

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

In this article we further advance in the theory of singular fully nonlinear operators modeled on the q-laplacian proving a Harnack inequality. We provide also several applications of this inequality and the ideas used for proving it. In doing so we have left various open questions, all of them related to the fact that the operator is not sub-linear.

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. Astarita G., Marrucci G.: Principles of Non-Newtonian Fluid Mechanics. McGraw-Hill, London (1974)

    Google Scholar 

  2. Birindelli, I., Demengel, F.: Comparison principle and Liouville type results for singular fully nonlinear operators. Ann. Fac. Sci. Toulouse Math. 6(13) N.2, 261–287 (2004)

    Google Scholar 

  3. Birindelli I., Demengel F.: First Eigenvalue and Maximun principle for fully nonlinear singular operators. Adv Partial Differ. Equ. 11(1), 91–119 (2006)

    MATH  MathSciNet  Google Scholar 

  4. Birindelli I., Demengel F.: Eigenvalue, maximun principle and regularity for fully non linear homogeneous operators. Commun. Pure Appl. Anal. 6(2), 335–366 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Birindelli, I., Demengel, F.: The Dirichlet problem for singular fully nonlinear operators. Discret. Contin. Dynam Syst. Suppl. 110–121 (2007)

  6. Birindelli I., Demengel F.: Eigenvalue and Dirichlet problem for fully-nonlinear operators in non-smooth domains. J. Math. Anal. Appl. 352(2), 822–835 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Birindelli, I., Demengel, F.: Eigenfunctions for singular fully nonlinear equations in unbounded regular domain. Preprint

  8. Birindelli I., Demengel F.: Eigenvalue, maximun principle and regularity for fully non linear homogeneous operators. Commun. Pure Appl. Anal. 6(2), 335–366 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Brezis H.: Semilinear equations in \({\mathbb{R}^{n}}\) without condition at infinity. Appl. Math. Optim. 12, 271–282 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  10. Caffarelli L.: Interior a priori estimates for solutions of fully non-linear equations. Ann. Math. 130, 189–213 (1989)

    Article  MathSciNet  Google Scholar 

  11. Caffarelli, L., Cabré, X.: Fully nonlinear elliptic equations. American Mathematical Society, Colloquium Publication, No. 43 (1995)

  12. Caffarelli L., Crandall M., Kocan M., Świech A.: On viscosity solutions of fully nonlinear equations with measurable ingredients. Commun. Pure Appl. Math. 49(4), 365–398 (1996)

    Article  MATH  Google Scholar 

  13. Cerutti C., Escauriaza L., Fabes E.: Uniqueness for some diffusions with discontinuous coefficients. Ann. Probab. 19(2), 525–537 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  14. Chen Y.G., Giga Y., Goto S.: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differ. Geom. 33, 749–786 (1991)

    MATH  MathSciNet  Google Scholar 

  15. Crandall M., Ishii H., Lions P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. AMS 27(1), 1–67 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  16. Dávila, G., Felmer, P., Quaas, A.: Alexandroff–Bakelman–Pucci estimate for singular or degenerate fully nonlinear elliptic equations. Comptes Rendus Mathmatique 347(19–20), 1165–1168 (2009)

    Google Scholar 

  17. Del Pino M., Letelier R.: The influence of domain geometry in boundary blow-up elliptic problems. Nonlinear Anal. Theory Methods Appl. 48(6), 897–904 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Delarue, F.: Krylov and Safonov estimates for degenerate quasilinear elliptic PDEs. Preprint

  19. Díaz G.: A maximum principle for fully nonlinear elliptic, eventually degenerate, second order equations in the whole space. Houston J. Math. 21(3), 507–524 (1995)

    MATH  MathSciNet  Google Scholar 

  20. Díaz G., Letelier R.: Explosive solutions of quasilinear elliptic equations: existence and uniqueness. Nonlinear Anal. Theory Methods Appl. 20(2), 97–125 (1993)

    Article  MATH  Google Scholar 

  21. Esteban M., Felmer P., Quaas A.: Super-linear elliptic equation for fully nonlinear operators without growth restrictions for the data. Proc. Roy. Soc. Edinburgh 53, 125–141 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  22. Evans C., Spruck J.: Motion of level sets by mean curvature. J. Differ. Geom. 33, 635–681 (1991)

    MATH  MathSciNet  Google Scholar 

  23. Giga Y., Goto S., Ishii H., Sato M-H.: Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ. Math. J. 40(2), 443–470 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  24. Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, New York (1983)

    MATH  Google Scholar 

  25. Imbert, C.: Alexandroff–Bakelman–Pucci estimate and Harnack inequality for degenerate fully non-linear elliptic equations. J. Differ. Equ., to appear

  26. Ishii H.: Viscosity solutions of non-linear partial differential equations. Sugaku Expo. 9, 135–152 (1996)

    Google Scholar 

  27. Junges Miotto, T.: The Aleksandrov–Bakelman–Pucci estimate for singular fully nonlinear operators. Preprint

  28. Juutinen P., Lindquist P., Manfredi J.: On the equivalence of viscosity solutions and weak solutions for a quasi linear equation. SIAM J. Math. Anal. 33(3), 699–717 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  29. Kassmann, M.: Harnack inequalities: an introduction. Bound. Value Prob. 2007, Article ID 81415, 21 pages (2007).

  30. Kato T.: Schrödinger operators with singular potentials. Israel J. Math. 13, 135–148 (1972)

    Article  MathSciNet  Google Scholar 

  31. Keller J.B.: On solutions of Δu =  f (u). Comm. Pure Appl. Math. 10, 503–510 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  32. Kondratev V., Nikishkin V.: Asymptotics near the boundary of a solution of singular boundary problem value problems for semilinear elliptic equations. Differ. Equ. 26, 345–348 (1990)

    MathSciNet  Google Scholar 

  33. Krylov, N.V., Safonov, M.V.: An estimate of the probability that a diffusion process hits a set of positive measure. Dokl. Akad. Nauk. SSSR 245, 253–255 (1979) (in Russian); Soviet Math. Dokl. 20 (1979), 253–255 (Engl. Transl.)

    Google Scholar 

  34. Loewner, C., Nirenberg, L.: Partial differential equations invariant under conformal projective transformations. Contributions to Analysis (a collection of papers dedicated to Lipman Bers), pp. 245–272. Academic Press, New York (1974)

  35. Moser J.: On Harnack’s theorem for elliptic differential equations. Commun. Pure Appl. Math. 14, 577–591 (1961)

    Article  MATH  Google Scholar 

  36. Ohnuma M., Sato M.-H.: Singular degenerate parabolic equations with applications to geometric evolutions. Differ. Int. Equ. 6(6), 1265–1280 (1993)

    MATH  MathSciNet  Google Scholar 

  37. Patrizi S.: The Neumann problem for singular fully nonlinear operators. J. Math. Pure Appl. 90(3), 286–311 (2008)

    MATH  MathSciNet  Google Scholar 

  38. Serrin J.: Local behavior of solution of quasi-linear elliptic equations. Acta Math. 111, 247–302 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  39. Trudinger N.S.: Harnack inequalities for nonuniformly elliptic divergence structure equations. Invent. Math. 64, 517–531 (1981)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, University of Texas, Austin, TX, USA

    Gonzalo Dávila

  2. Departamento de Ingeniería Matemática, Centro de Modelamiento Matemático, UMR2071 CNRS-UChile, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

    Patricio Felmer

  3. Departamento de Matemática, Universidad Técnica Santa María, Casilla 110, Avda. España 1680, Valparaíso, Chile

    Alexander Quaas

Authors
  1. Gonzalo Dávila
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Patricio Felmer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexander Quaas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gonzalo Dávila.

Additional information

Communicated by L. C. Evans.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dávila, G., Felmer, P. & Quaas, A. Harnack inequality for singular fully nonlinear operators and some existence results. Calc. Var. 39, 557–578 (2010). https://doi.org/10.1007/s00526-010-0325-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-010-0325-3

Mathematics Subject Classification (2000)

Search

Navigation