Open Access

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

Boundary Value Problems20072007:078029

DOI: 10.1155/2007/78029

Received: 27 June 2006

Accepted: 27 October 2006

Published: 17 January 2007

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.

[123456789101112131415]

Authors’ Affiliations

(1)
Department of Mathematics, Western Kentucky University

References

  1. Bhattacharya T:On the properties of https://static-content.springer.com/image/art%3A10.1155%2F2007%2F78029/MediaObjects/13661_2006_Article_664_IEq1_HTML.gif -harmonic functions and an application to capacitary convex rings. Electronic Journal of Differential Equations 2002,2002(101):1-22.
  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 Article
  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 Article
  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 Article
  5. Manfredi JJ, Weitsman A:On the Fatou theorem for https://static-content.springer.com/image/art%3A10.1155%2F2007%2F78029/MediaObjects/13661_2006_Article_664_IEq2_HTML.gif -harmonic functions. Communications in Partial Differential Equations 1988,13(6):651-668. 10.1080/03605308808820556MATHMathSciNetView Article
  6. Aronsson G:Construction of singular solutions to the https://static-content.springer.com/image/art%3A10.1155%2F2007%2F78029/MediaObjects/13661_2006_Article_664_IEq3_HTML.gif -harmonic equation and its limit equation for https://static-content.springer.com/image/art%3A10.1155%2F2007%2F78029/MediaObjects/13661_2006_Article_664_IEq4_HTML.gif . Manuscripta Mathematica 1986,56(2):135-158. 10.1007/BF01172152MathSciNetView Article
  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 Article
  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 Article
  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.MATHMathSciNet
  10. Bhattacharya T, DiBenedetto E, Manfredi JJ:Limits as https://static-content.springer.com/image/art%3A10.1155%2F2007%2F78029/MediaObjects/13661_2006_Article_664_IEq5_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2007%2F78029/MediaObjects/13661_2006_Article_664_IEq6_HTML.gif 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. Bhattacharya T:On the behaviour of https://static-content.springer.com/image/art%3A10.1155%2F2007%2F78029/MediaObjects/13661_2006_Article_664_IEq7_HTML.gif -harmonic functions near isolated points. Nonlinear Analysis. Theory, Methods & Applications 2004,58(3-4):333-349. 10.1016/j.na.2004.02.028MATHView Article
  12. Bhattacharya T:On the behaviour of https://static-content.springer.com/image/art%3A10.1155%2F2007%2F78029/MediaObjects/13661_2006_Article_664_IEq8_HTML.gif -harmonic functions on some special unbounded domains. Pacific Journal of Mathematics 2005,219(2):237-253. 10.2140/pjm.2005.219.237MATHMathSciNetView Article
  13. Lindqvist P, Manfredi JJ:The Harnack inequality for https://static-content.springer.com/image/art%3A10.1155%2F2007%2F78029/MediaObjects/13661_2006_Article_664_IEq9_HTML.gif -harmonic functions. Electronic Journal of Differential Equations 1995,1995(4):1-5.MathSciNet
  14. Savin O: https://static-content.springer.com/image/art%3A10.1155%2F2007%2F78029/MediaObjects/13661_2006_Article_664_IEq10_HTML.gif 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 Article
  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 Article

Copyright

© 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.