International Mathematics Research Notices

International Mathematics Research Notices (2008) Vol. 2008 : article ID rnm163, 23 pages, doi:10.1093/imrn/rnm163 published on February 6, 2008

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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Asymptotic Behavior of Infinity Harmonic Functions Near an Isolated Singularity

Ovidiu Savin¹, Changyou Wang² and Yifeng Yu³

¹ Department of Mathematics, Columbia University, New York, NY 10027 USA
² Department of Mathematics, University of Kentucky, Lexington, KY 40513 USA
³ Department of Mathematics, University of Texas at Austin, Austin, TX 78712, USA

Correspondence: Correspondence to be sent to: savin{at}math.columbia.edu

In this paper, we prove that if n 2 and x₀ is an isolated singularity of a non-negative infinity harmonic function u, then either x₀ is a removable singularity of u or u(x)=u(x₀)+c|x–x₀|+o(|x–x₀|) near x₀ for some fixed constant c0. In particular, if x₀ is nonremovable, then u has a local maximum or a local minimum at x₀. We also prove a Bernstein-type theorem, which asserts that if u is a uniformly Lipschitz continuous, one-side bounded infinity harmonic function in then it must be a cone function with center at 0.

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This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive Download to citation manager Request Permissions How to cite this article Google Scholar Articles by Savin, O. Articles by Yu, Y. Search for Related Content Social Bookmarking          What's this?