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

Boundary Value Problems volume 2007, Article number: 078029 (2007) Cite this article

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Abstract

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.

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References

  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/j.na.2004.02.028

  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

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

    • Tilak Bhattacharya

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Correspondence to Tilak Bhattacharya.

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

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

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Keywords

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