Skip to main content


A Boundary Harnack Principle for Infinity-Laplacian and Some Related Results

Article metrics

  • 1086 Accesses

  • 1 Citations


We prove a boundary comparison principle for positive infinity-harmonic functions for smooth boundaries. As consequences, we obtain (a) a doubling property for certain positive infinity-harmonic functions in smooth bounded domains and the half-space, and (b) the optimality of blowup rates of Aronsson's examples of singular solutions in cones.



  1. 1.

    Bhattacharya T:On the properties of-harmonic functions and an application to capacitary convex rings. Electronic Journal of Differential Equations 2002,2002(101):1-22.

  2. 2.

    Bauman P: Positive solutions of elliptic equations in nondivergence form and their adjoints. Arkiv för Matematik 1984,22(2):153-173. 10.1007/BF02384378

  3. 3.

    Caffarelli L, Fabes E, Mortola S, Salsa S: Boundary behavior of nonnegative solutions of elliptic operators in divergence form. Indiana University Mathematics Journal 1981,30(4):621-640. 10.1512/iumj.1981.30.30049

  4. 4.

    Fabes E, Garofalo N, Marín-Malave S, Salsa S: Fatou theorems for some nonlinear elliptic equations. Revista Matemática Iberoamericana 1988,4(2):227-251.

  5. 5.

    Manfredi JJ, Weitsman A:On the Fatou theorem for-harmonic functions. Communications in Partial Differential Equations 1988,13(6):651-668. 10.1080/03605308808820556

  6. 6.

    Aronsson G:Construction of singular solutions to the-harmonic equation and its limit equation for. Manuscripta Mathematica 1986,56(2):135-158. 10.1007/BF01172152

  7. 7.

    Bhattacharya T: A note on non-negative singular infinity-harmonic functions in the half-space. Revista Matemática Complutense 2005,18(2):377-385.

  8. 8.

    Aronsson G, Crandall MG, Juutinen P: A tour of the theory of absolutely minimizing functions. Bulletin of the American Mathematical Society. New Series 2004,41(4):439-505. 10.1090/S0273-0979-04-01035-3

  9. 9.

    Crandall MG, Evans LC, Gariepy RF: Optimal Lipschitz extensions and the infinity Laplacian. Calculus of Variations and Partial Differential Equations 2001,13(2):123-139.

  10. 10.

    Bhattacharya T, DiBenedetto E, Manfredi JJ:Limits as of and related extremal problems. Some topics in nonlinear PDEs (Turin, 1989). Rendiconti del Seminario Matematico. Università e Politecnico di Torino 1989, 15-68 (1991). special issue

  11. 11.

    Bhattacharya T:On the behaviour of-harmonic functions near isolated points. Nonlinear Analysis. Theory, Methods & Applications 2004,58(3-4):333-349. 10.1016/

  12. 12.

    Bhattacharya T:On the behaviour of-harmonic functions on some special unbounded domains. Pacific Journal of Mathematics 2005,219(2):237-253. 10.2140/pjm.2005.219.237

  13. 13.

    Lindqvist P, Manfredi JJ:The Harnack inequality for-harmonic functions. Electronic Journal of Differential Equations 1995,1995(4):1-5.

  14. 14.

    Savin O: regularity for infinity harmonic functions in two dimensions. Archive for Rational Mechanics and Analysis 2005,176(3):351-361. 10.1007/s00205-005-0355-8

  15. 15.

    Barles G, Busca J: Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term. Communications in Partial Differential Equations 2001,26(11-12):2323-2337. 10.1081/PDE-100107824

Download references

Author information

Correspondence to Tilak Bhattacharya.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Bhattacharya, T. A Boundary Harnack Principle for Infinity-Laplacian and Some Related Results. Bound Value Probl 2007, 078029 (2007) doi:10.1155/2007/78029

Download citation


  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Equation
  • Related Result