Uniqueness of the Boundary Value Problem of Harmonic Maps via Harmonic Boundary

Abstract

We prove the uniqueness of solutions for the boundary value problem of harmonic maps in the setting: given any continuous data f on the harmonic boundary of a complete Riemannian manifold with image within a regular geodesic ball, there exists a unique harmonic map, which is a limit of a sequence of harmonic maps with finite total energy in the sense of the supremum norm, from the manifold into the ball taking the same boundary value at each harmonic boundary point as that of f.

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Correspondence to Yong Hah Lee.

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This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2017R1D1A1B04030647).

Communicated by See Keong Lee.

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Lee, Y.H. Uniqueness of the Boundary Value Problem of Harmonic Maps via Harmonic Boundary. Bull. Malays. Math. Sci. Soc. 43, 2733–2743 (2020). https://doi.org/10.1007/s40840-019-00830-9

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Keywords

  • Harmonic map
  • Harmonic boundary
  • Boundary value problem
  • Uniqueness

Mathematics Subject Classification

  • 58E20
  • 53C43