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Archive for Rational Mechanics and Analysis

, Volume 180, Issue 1, pp 75–95 | Cite as

Regularity of the Inverse of a Planar Sobolev Homeomorphism

  • Stanislav Hencl
  • Pekka KoskelaEmail author
Article

Abstract

Let Open image in new window be a domain. Suppose that fW1,1loc(Ω,R2) is a homeomorphism such that Df(x) vanishes almost everywhere in the zero set of J f . We show that f-1W1,1loc(f(Ω),R2) and that Df−1(y) vanishes almost everywhere in the zero set of Open image in new window Sharp conditions to quarantee that f−1W1, q (f(Ω),R2) for some 1<q≤2 are also given.

Keywords

Neural Network Complex System Nonlinear Dynamics Electromagnetism Sharp Condition 

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References

  1. 1.
    Astala, K., Iwaniec, T., Martin, G., Onninen, J.: Extremal mappings of finite distortion. To appear in Proc. London Math. Soc. Google Scholar
  2. 2.
    Ball, J.: Global invertibility of Sobolev functions and the interpenetration of matter. Proc. Roy. Soc. Edinburgh Sect. A 88, 315–328 (1981)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Brakalova, M.A., Jenkins, J.A.: On solutions of the Beltrami equation. J. Anal. Math. 76, 67–92 (1998)CrossRefMathSciNetGoogle Scholar
  4. 4.
    David, G.: Solutions de l'equation de Beltrami avec ||μ||=1. Ann. Acad. Sci. Fenn. Ser. A I, Math. 13, 25–70 (1988)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Dellacherie, C., Meyer, P.A.: Probabilities and potential. North-Holland Mathematics Studies, 29, North-Holland Publishing Co. 1978Google Scholar
  6. 6.
    Faraco, D., Koskela, P., Zhong, X.: Mappings of finite distortion: The degree of regularity. Adv. Math 190, 300–318 (2005)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Federer, H.: Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag, New York, 1969 (Second edition 1996)Google Scholar
  8. 8.
    Fonseca, I., Gangbo, W.: Degree Theory in Analysis and Applications. Clarendon Press, Oxford, 1995Google Scholar
  9. 9.
    Gehring, F.W., Lehto, O.: On the total differentiability of functions of a complex variable. Ann. Acad. Sci. Fenn. Ser. A I 272, 1–9 (1959)Google Scholar
  10. 10.
    Gehring, F.W., Väisälä, J.: Hausdorff dimension and quasiconformal mappings. J. London Math. Soc. 6, 504–512 (1973)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Gutlyanskii, V., Martio, O., Sugawa, T., Vuorinen, M.: On the Degenerate Beltrami Equation. Trans. Amer. Math. Soc 357, 875–900 (2005)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Heinonen, J., Koskela, P.: Sobolev mappings with integrable dilatations. Arch. Ration. Mech. Anal. 125, 81–97 (1993)CrossRefGoogle Scholar
  13. 13.
    Hencl, S., Koskela, P., Malý, J.: Regularity of the inverse of a Sobolev homeomorphism in space. In preparation.Google Scholar
  14. 14.
    Hencl, S., Koskela, P., Onninen, J.: A note on extremal mappings of finite distortion. Math. Res. Lett. 12, 231–238 (2005)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Iwaniec, T., Martin, G.: Geometric function theory and nonlinear analysis. Oxford Mathematical Monographs, Clarendon Press, Oxford, 2001Google Scholar
  16. 16.
    Iwaniec, T., Martin, G.: Beltrami equation. To appear in Mem. Amer. Math. Soc. Google Scholar
  17. 17.
    Iwaniec, T., Šverák, V.: On mappings with integrable dilatation. Proc. Amer. Math. Soc. 118, 181–188 (1993)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Kauhanen, J.: An example concerning the zero set of the Jacobian. To appear in J. Math. Anal. Appl. Google Scholar
  19. 19.
    Kauhanen, J., Koskela, P., Malý, J.: Mappings of finite distortion: Condition N. Michigan Math. J. 49, 169–181 (2001)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Kauhanen, J., Koskela, P., Malý, J.: Mappings of finite distortion: Discreteness and openness. Arch. Ration. Mech. Anal. 160, 135–151 (2001)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Kauhanen, J., Koskela, P., Malý, J., Onninen, J., Zhong, X.: Mappings of finite distortion: Sharp Orlicz-conditions. Rev. Mat. Iberoamericana 19, 857–872 (2003)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Koskela, P., Malý, J.: Mappings of finite distortion: the zero set of the Jacobian. J. Eur. Math. Soc. 5, 95–105 (2003)CrossRefGoogle Scholar
  23. 23.
    Koskela, P., Onninen, J.: Mappings of finite distortion: Capacity and modulus inequalities. To appear in J. Reine Angew. Math. Google Scholar
  24. 24.
    Malý, J.: Lectures on change of variables in integral. Preprint 305, Department of Math., University of Helsinki (2001)Google Scholar
  25. 25.
    Moscariello, G.: On the integrability of the Jacobian in Orlicz spaces. Math. Japanica 40, 323–329 (1992)Google Scholar
  26. 26.
    Müller, S.: Higher integrability of determinants and weak convergence in L 1. J. Reine Angew. Math. 412, 20–34 (1990)MathSciNetGoogle Scholar
  27. 27.
    Müller, S., Tang, Q., Yan, B.S.: On a new class of elastic deformations not allowing for cavitation. Ann. Inst. H. Poincaré Anal. Non Lináire 11, 217–243 (1994)ADSCrossRefGoogle Scholar
  28. 28.
    Ponomarev, S.: Examples of homeomorphisms in the class ACTLp which do not satisfy the absolute continuity condition of Banach. Dokl. Akad. Nauk USSR201, 1053–1054 (1971)Google Scholar
  29. 29.
    Rickman, S.: Quasiregular mappings. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 26. Springer-Verlag, Berlin, 1993Google Scholar
  30. 30.
    Ryazanov, V., Srebro, U., Yakubov, E.: BMO-Quasiconformal mappings. J. Analyse Math. 83, 1–20 (2001)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Tang, Q.: Almost-everywhere injectivity in nonlinear elasticity. Proc. Roy. Soc. Edinburgh Sect. A 109, 79–95 (1988)CrossRefMathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of JyväskyläJyväskyläFinland

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