\((n,\rho )\)-harmonic mappings and energy minimal deformations between annuli

  • David KalajEmail author


We extend the main results obtained by Iwaniec and Onninen in Memoirs of the AMS (2012). In this paper, we solve the \((\rho ,n)\)-energy minimization problem for Sobolev homeomorphisms between two concentric annuli in the Euclidean space \(\mathbf {R}^n\). Here \(\rho \) is a radial metric defined in the image annulus. The key element in the proofs is the solution to the Euler–Lagrange equation for a radial harmonic mapping. This is a new contribution on the topic related to the famous J. C. C. Nitsche conjecture on harmonic mappings between annuli on the complex plane. Namely we prove that the minimum of \((\rho ,n)\)-energy of diffeomorphisms between annuli is attained by a certain \((\rho ,n)\)-harmonic diffeomorphisms if and only if the original annulus can be mapped onto the image annulus by a radial \((\rho ,n)\)-harmonic diffeomorphisms and the last fact is equivalent with a certain inequality for annuli which we call a generalized J. C. C. Nitsche type inequality.

Mathematics Subject Classification

Primary 31A05 Secondary 42B30 



I am grateful to the referee for many useful suggestions and corrections.


  1. 1.
    Astala, K., Iwaniec, T., Martin, G.: Deformations of annuli with smallest mean distortion. Arch. Ration. Mech. Anal. 195, 899–921 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Csörnyei, M., Hencl, S., Malý, J.: Homeomorphisms in the Sobolev space \(\mathscr {W}^{1, n-1}\). J. Reine Angew. Math. 644, 221–235 (2010)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Hencl, S., Koskela, P.: Regularity of the inverse of a planar Sobolev homeomorphism. Arch. Ration. Mech. Anal. 180, 75–95 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Iwaniec, T., Kovalev, L.V., Onninen, J.: The Nitsche conjecture. J. Am. Math. Soc. 24(2), 345–373 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Iwaniec, T., Koh, N.-T., Kovalev, L.V., Onninen, J.: Existence of energy-minimal diffeomorphisms between doubly connected domains. Invent. Math. 186(3), 667–707 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Iwaniec, T., Onninen, J.: \(n\)-harmonic mappings between annuli: the art of integrating free Lagrangians. Mem. Am. Math. Soc. 218(1023), viii+105 (2012)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Jost, J., Li-Jost, X.: Calculus of Variations. Cambridge Studies in Advanced Mathematics, vol. 64. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  8. 8.
    Kalaj, D.: Energy-minimal diffeomorphisms between doubly connected Riemann surfaces. Calc. Var. Partial Differ. Equ. 51(1–2), 465–494 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kalaj, D.: Deformations of annuli on Riemann surfaces and the generalization of Nitsche conjecture. J. Lond. Math. Soc. (2) 93(3), 683–702 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kalaj, D.: On the Nitsche conjecture for harmonic mappings in \(\mathbb{R}^2\) and \(\mathbb{R}^3\). Israel J. Math. 150, 241–251 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kalaj, D.: On J. C. C. Nitsche type inequality for annuli on Riemann surfaces. Israel J. Math. 218, 67–281 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lyzzaik, A.: The modulus of the image annuli under univalent harmonic mappings and a conjecture of J.C.C. Nitsche. J. Lond. Math. Soc. 64, 369–384 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Marković, V.: Harmonic maps and the Schoen conjecture. J. Am. Math. Soc. 30, 799–817 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Nitsche, J.C.C.: On the modulus of doubly connected regions under harmonic mappings. Am. Math. Mon. 69, 781–782 (1962)CrossRefGoogle Scholar
  15. 15.
    Weitsman, A.: Univalent harmonic mappings of annuli and a conjecture of J.C.C. Nitsche. Israel J. Math. 124, 327–331 (2001)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesHuaqiao UniversityQuanzhouPeople’s Republic of China
  2. 2.Faculty of Natural Sciences and MathematicsUniversity of MontenegroPodgoricaMontenegro

Personalised recommendations