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A Liouville theorem for infinity harmonic functions

  • Guanghao Hong
  • Yizhen Zhao
Article
  • 14 Downloads

Abstract

We proved that a Lipschitz entire infinity harmonic function on \(\mathbb {R}^n\) must asymptotically tend to a plane at infinity and it must be a plane provided the length of its gradient is continuous at infinity. We also prove that if the length of the gradient of an infinity harmonic function is continuous at some interior point then the gradient of the function is continuous at this point. All the proofs of the above conclusions are based on some modifications of Evans-Smart’s argument.

Mathematics Subject Classification

Primary 35J15 35J60 35J70 Secondary 49N60 

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Notes

Acknowledgments

This work is supported by the National Nature Science Foundation of China: NSFC 11301411 and 11671316. The first author would like to thank Professor Dongsheng Li for many helpful conversations and encouragement.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXi’an Jiaotong UniversityXi’anPeople’s Republic of China

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