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Failure of strong unique continuation for harmonic functions on RCD spaces

  • Qin Deng and Xinrui Zhao EMAIL logo
Published/Copyright: December 15, 2022

Abstract

Unique continuation of harmonic functions on RCD space is a long-standing open problem, with little known even in the setting of Alexandrov spaces. In this paper, we establish the weak unique continuation theorem for harmonic functions on RCD ( K , 2 ) spaces and give a counterexample for strong unique continuation in the setting of RCD ( K , N ) space for any N 4 and any K .

Award Identifier / Grant number: DMS-1812142

Funding statement: Xinrui Zhao is supported by the National Science Foundation, Grant DMS-1812142.

Acknowledgements

We are very grateful to Professor Tobias Colding for introducing us to the unique continuation problem and several inspirational discussions. We thank Professor Vitali Kapovitch for helpful discussions and comments during the writing of this paper. We also thank the anonymous referees for comments and suggestions.

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Received: 2022-05-11
Revised: 2022-11-02
Published Online: 2022-12-15
Published in Print: 2023-02-01

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