Abstract
This note deals with the existence and uniqueness of a minimiser of the following Grötzsch-type problem \(\mathop {\inf }\limits_{f \in \mathcal{F}} \iint_{Q_1 } {\phi (K(z,f))\lambda (x)dxdy}\) under some mild conditions, where F denotes the set of all homeomorphims f with finite linear distortion K(z, f) between two rectangles Q1 and Q2 taking vertices into vertices, ϕ is a positive, increasing and convex function, and λ is a positive weight function. A similar problem of Nitsche-type, which concerns the minimiser of some weighted functional for mappings between two annuli, is also discussed. As by-products, our discussion gives a unified approach to some known results in the literature concerning the weighted Grötzsch and Nitsche problems.
Similar content being viewed by others
References
Astala K, Iwaniec T, Martin G J. Elliptic Partial Diffenrential Equations and Quasiconformal Mappings in the Plane. Princeton: Princeton University Press, 2009
Astala K, Iwaniec T, Martin G J. Deformations of annuli with smallest mean distortion. Arch Ration Mech Anal, 2010; 195: 899–921
Astala K, Iwaniec T, Martin G J, et al. Extremal mappings of finite distortion. Proc Lond Math Soc, 2005; 91: 655–702
Grötzsch H. Über die Verzerrung bei schlichten nichtkonformen Abbildungen und über eine damit zusammenhängende Erweiterung des Picardschen Sates. Ber Verh Sächs Akad Wiss Leipzig, 1928; 80: 503–507
Iwaniec T, Kovalev L V, Onninen J. The Nitsche conjecture. J Amer Math Soc, 2011; 24: 345–373
Iwaniec T, Martin G J. Geometric Function Theory and Non-linear Analysis. Oxford: Oxford University Press, 2001
Iwaniec T, Martin G J, Onninen J. On minimiser of L p-mean distortion. Comput Methods Funct Theory, 2014; 14: 399–416
Kalaj D. Harmonic maps between annuli on Riemann surfaces. Israel J Math, 2011; 182: 123–147
Kalaj D. Deformations of annuli on Riemann surfaces with smallest mean distortion. Arxiv:1005.5269, 2010
Martin G J. The Teichmüller problem for mean distortion. Ann Acad Sci Fenn A IMath, 2009; 34: 233–247
Martin G J, Mckubre-Jordens M. Deformation with smallest weighted L p average distortion and Nitsche-type phenomena. J Lond Math Soc, 2012; 85: 282–300
Nitsche J C C. On the modulus of doubly connected regions under harmonic mappings. Amer Math Monthly, 1962; 69: 781–782
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Feng, X., Tang, S., Wu, C. et al. A unified approach to the weighted Grötzsch and Nitsche problems for mappings of finite distortion. Sci. China Math. 59, 673–686 (2016). https://doi.org/10.1007/s11425-015-5078-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-015-5078-1