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

  • Yong Hah LeeEmail author


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.


Harmonic map Harmonic boundary Boundary value problem Uniqueness 

Mathematics Subject Classification

58E20 53C43 



  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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Giaquinta, M., Hildebrandt, S.: A priori estimates for harmonic mappings. J. Reine Angew. Math. 336, 124–164 (1982)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Hildebrandt, S., Kaul, H., Widman, K.: An existence theorem for harmonic mappings of Riemannian manifolds. Acta Math. 138, 1–16 (1977)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Lee, Y.H.: Rough isometry and energy finite solutions of elliptic equations on Riemannian manifolds. Math. Ann. 318, 181–204 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Lee, Y.H.: Asymptotic boundary value problem of harmonic maps via harmonic boundary. Potential Anal. 41, 463–468 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Lee, Y.H.: Royden decomposition for harmonic maps with finite total energy. Results Math. 71, 687–692 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Sario, L., Nakai, M.: Classification Theory of Riemann Surfaces. Springer, Berlin (1970)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  1. 1.Department of Mathematics EducationEwha Womans UniversitySeoulKorea

Personalised recommendations