Abstract
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.
Similar content being viewed by others
References
Astala, K., Iwaniec, T., Martin, G.: Deformations of annuli with smallest mean distortion. Arch. Ration. Mech. Anal. 195, 899–921 (2010)
Csörnyei, M., Hencl, S., Malý, J.: Homeomorphisms in the Sobolev space \(\mathscr {W}^{1, n-1}\). J. Reine Angew. Math. 644, 221–235 (2010)
Hencl, S., Koskela, P.: Regularity of the inverse of a planar Sobolev homeomorphism. Arch. Ration. Mech. Anal. 180, 75–95 (2006)
Iwaniec, T., Kovalev, L.V., Onninen, J.: The Nitsche conjecture. J. Am. Math. Soc. 24(2), 345–373 (2011)
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)
Iwaniec, T., Onninen, J.: \(n\)-harmonic mappings between annuli: the art of integrating free Lagrangians. Mem. Am. Math. Soc. 218(1023), viii+105 (2012)
Jost, J., Li-Jost, X.: Calculus of Variations. Cambridge Studies in Advanced Mathematics, vol. 64. Cambridge University Press, Cambridge (1998)
Kalaj, D.: Energy-minimal diffeomorphisms between doubly connected Riemann surfaces. Calc. Var. Partial Differ. Equ. 51(1–2), 465–494 (2014)
Kalaj, D.: Deformations of annuli on Riemann surfaces and the generalization of Nitsche conjecture. J. Lond. Math. Soc. (2) 93(3), 683–702 (2016)
Kalaj, D.: On the Nitsche conjecture for harmonic mappings in \(\mathbb{R}^2\) and \(\mathbb{R}^3\). Israel J. Math. 150, 241–251 (2005)
Kalaj, D.: On J. C. C. Nitsche type inequality for annuli on Riemann surfaces. Israel J. Math. 218, 67–281 (2017)
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)
Marković, V.: Harmonic maps and the Schoen conjecture. J. Am. Math. Soc. 30, 799–817 (2017)
Nitsche, J.C.C.: On the modulus of doubly connected regions under harmonic mappings. Am. Math. Mon. 69, 781–782 (1962)
Weitsman, A.: Univalent harmonic mappings of annuli and a conjecture of J.C.C. Nitsche. Israel J. Math. 124, 327–331 (2001)
Acknowledgements
I am grateful to the referee for many useful suggestions and corrections.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Malchiodi.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kalaj, D. \((n,\rho )\)-harmonic mappings and energy minimal deformations between annuli. Calc. Var. 58, 51 (2019). https://doi.org/10.1007/s00526-019-1490-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-019-1490-7