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Deformations of Annuli with Smallest Mean Distortion

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Abstract

We determine the extremal mappings with smallest mean distortion for mappings of annuli. As a corollary, we find that the Nitsche harmonic maps are Dirichlet energy minimizers among all homeomorphisms \({h:{{\mathbb A}}(r, R) \to {{\mathbb A}}(r', R')}\) . However, outside the Nitsche range of the modulus of the annuli, within the class of homeomorphisms, no such energy minimizers exist. In this case we identify the BV-limits of minimizers.

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References

  1. Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, New York (2000)

    MATH  Google Scholar 

  2. Astala, K., Iwaniec, T., Martin, G.J.: In preparation

  3. Astala K., Iwaniec T., Martin G.J.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton University Press, Princeton (2009)

    MATH  Google Scholar 

  4. Astala K., Iwaniec T., Martin G., Onninen J.: Extremal mappings of finite distortion. Proc. London Math. Soc. 91(3), 655–702 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63(4), 337–403 (1976/77)

    Google Scholar 

  6. Duren P.: Harmonic Mappings. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  7. Federer, H.: Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band, Vol. 153. Springer-Verlag New York Inc., New York, 2nd edn, 1996

  8. Giaquinta M., Modica G., Soucek J.: Area and the area formula. Rend. Sem. Mat. Fis. Milano 62, 53–87 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hajlasz P.: Change of variables formula under minimal assumptions. Colloq. Math. 64(1), 93–101 (1993)

    MATH  MathSciNet  Google Scholar 

  10. Hencl S., Koskela P.: Regularity of the inverse of a planar Sobolev homeomorphism. Arch. Ration. Mech. Anal. 180, 75–95 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hencl S., Koskela P., Maly J.: Regularity of the inverse of a Sobolev homeomorphism in space. Proc. R. Soc. Edinburgh Sect. A 136, 1267–1285 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hencl S., Koskela P., Onninen J.: A note on extremal mappings of finite distortion. Math. Res. Lett. 12, 231–237 (2005)

    MATH  MathSciNet  Google Scholar 

  13. Iwaniec T., Martin G.J.: Geometric Function Theory and Nonlinear Analysis. Oxford University Press, New York (2001)

    MATH  Google Scholar 

  14. Iwaniec T., Martin G.J.: The Beltrami equation. Memoirs Amer. Math. Soc. 191, 1–92 (2008)

    MathSciNet  Google Scholar 

  15. Jordens, M., Martin, G.J.: Deformations with Smallest Weighted Mean Distortion (to appear)

  16. Kalaj D.: On the Nitsche conjecture for harmonic mappings in \({{\mathbb R}^{2}}\) and \({{\mathbb R}^{3}}\) . Israel J. Math. 150, 241–251 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  17. Lyzzaik A.: The modulus of the image annuli under univalent harmonic mappings and a conjecture of J.C.C. Nitsche. J. London Math. Soc. 64, 369–384 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. Lyzzaik A.: Univalent harmonic mappings of annuli. Publ. Inst. Math. (Beograd) (N.S.) 75(89), 173–183 (2004)

    Article  MathSciNet  Google Scholar 

  19. Martin G.J.: The Teichmüller problem for mean distortion. Ann. Acad. Sci. Fenn. 34, 233–247 (2009)

    MATH  ADS  Google Scholar 

  20. Nitsche J.C.C.: On the modulus of doubly connected regions under harmonic mappings. Am. Math. Monthly 69, 781–782 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  21. Schober, G.: Planar Harmonic Mappings. Computational Methods and Function Theory, (Valparaso, 1989), pp. 171–176, Lecture Notes in Math., vol. 1435. Springer, Berlin, 1990

  22. Weitsman A.: Univalent harmonic mappings of annuli and a conjecture of J.C.C. Nitsche. Israel J. Math. 124, 327–331 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Helsinki, Helsinki, Finland

    K. Astala

  2. Department of Mathematics, Syracuse University, Syracuse, USA

    T. Iwaniec

  3. Institute for Advanced Study, Massey University, Palmerston North, New Zealand

    G. Martin

Authors
  1. K. Astala
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  2. T. Iwaniec
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  3. G. Martin
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Corresponding author

Correspondence to G. Martin.

Additional information

Communicated by S. Müller

Research supported in part by grants from the Finnish Academy, the U.S. National Science Foundation and the N.Z. Marsden Fund.

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Astala, K., Iwaniec, T. & Martin, G. Deformations of Annuli with Smallest Mean Distortion. Arch Rational Mech Anal 195, 899–921 (2010). https://doi.org/10.1007/s00205-009-0231-z

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  • DOI: https://doi.org/10.1007/s00205-009-0231-z

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