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


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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  1. 1.

    Avilés, P., Choi, H.I., Micallef, M.: Boundary behavior of harmonic maps on non-smooth domains and complete negatively curved manifolds. J. Funct. Anal. 99, 293–331 (1991)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Giaquinta, M., Hildebrandt, S.: A priori estimates for harmonic mappings. J. Reine Angew. Math. 336, 124–164 (1982)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Hildebrandt, S., Kaul, H., Widman, K.: An existence theorem for harmonic mappings of Riemannian manifolds. Acta Math. 138, 1–16 (1977)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Kendall, W.S.: Probability, convexity, and harmonic maps with small image I: uniqueness and fine existence. Proc. Lond. Math. Soc. 61, 371–406 (1990)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Lee, Y.H.: Rough isometry and energy finite solutions of elliptic equations on Riemannian manifolds. Math. Ann. 318, 181–204 (2000)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Lee, Y.H.: Asymptotic boundary value problem of harmonic maps via harmonic boundary. Potential Anal. 41, 463–468 (2014)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Lee, Y.H.: Royden decomposition for harmonic maps with finite total energy. Results Math. 71, 687–692 (2017)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Sario, L., Nakai, M.: Classification Theory of Riemann Surfaces. Springer, Berlin (1970)

    Book  Google Scholar 

  9. 9.

    Sung, C.J., Tam, L.F., Wang, J.: Bounded harmonic maps on a class of manifolds. Proc. Am. Math. Soc. 124, 2241–2248 (1996)

    MathSciNet  Article  Google Scholar 

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

Mathematics Subject Classification

  • 58E20
  • 53C43