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  • Research Article
  • Open Access

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

Boundary Value Problems20072007:078029

  • Received: 27 June 2006
  • Accepted: 27 October 2006
  • Published:


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.


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


Authors’ Affiliations

Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101, USA


  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.Google Scholar
  2. Bauman P: Positive solutions of elliptic equations in nondivergence form and their adjoints. Arkiv för Matematik 1984,22(2):153-173. 10.1007/BF02384378MATHMathSciNetView ArticleGoogle Scholar
  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.1981.30.30049MATHMathSciNetView ArticleGoogle Scholar
  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.MATHView ArticleGoogle Scholar
  5. Manfredi JJ, Weitsman A:On the Fatou theorem for -harmonic functions. Communications in Partial Differential Equations 1988,13(6):651-668. 10.1080/03605308808820556MATHMathSciNetView ArticleGoogle Scholar
  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/BF01172152MathSciNetView ArticleGoogle Scholar
  7. Bhattacharya T: A note on non-negative singular infinity-harmonic functions in the half-space. Revista Matemática Complutense 2005,18(2):377-385.MATHView ArticleGoogle Scholar
  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/S0273-0979-04-01035-3MATHMathSciNetView ArticleGoogle Scholar
  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.MATHMathSciNetGoogle Scholar
  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 issueGoogle Scholar
  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/ ArticleGoogle Scholar
  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.2005.219.237MATHMathSciNetView ArticleGoogle Scholar
  13. Lindqvist P, Manfredi JJ:The Harnack inequality for -harmonic functions. Electronic Journal of Differential Equations 1995,1995(4):1-5.MathSciNetGoogle Scholar
  14. Savin O: regularity for infinity harmonic functions in two dimensions. Archive for Rational Mechanics and Analysis 2005,176(3):351-361. 10.1007/s00205-005-0355-8MATHMathSciNetView ArticleGoogle Scholar
  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-100107824MATHMathSciNetView ArticleGoogle Scholar


© Tilak Bhattacharya. 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.