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Close-to-harmonic extensions on the plane

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Abstract

In this paper, some explicit harmonic homeomorphic extensions are constructed. Necessary and sufficient conditions are obtained for the harmonic extensions to be quasiconformal. The extensions considered include those from the unit circle (or real axis) to the unit disk (or upper half plane) and from the unit disk (or upper half plane) to the whole plane. Furthermore, a new concept called close-to-harmonic extension is introduced and several open problems are proposed.

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Acknowledgements

The authors would like to thank the referee for a detailed report on the manuscript with several useful suggestions.

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Authors and Affiliations

  1. School of Mathematical Sciences, Anhui University, Hefei, 230601, People’s Republic of China

    Bo-Yong Long & Qi-Han Wang

  2. Department of Mathematics, Brigham Young University, Provo, UT, 84602, USA

    Michael Dorff

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  1. Bo-Yong Long
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  2. Qi-Han Wang
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  3. Michael Dorff
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Correspondence to Bo-Yong Long.

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Communicated by Adrian Constantin.

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Supported by Natural Science Foundation of Anhui Province(1908085MA18) and Foundations of Anhui Educational Committee (KJ2020A0002), China.

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Long, BY., Wang, QH. & Dorff, M. Close-to-harmonic extensions on the plane. Monatsh Math 197, 655–675 (2022). https://doi.org/10.1007/s00605-022-01668-3

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