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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 c!=0. 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 Formula then it must be a cone function with center at 0.


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