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Revista Matemática Complutense

, Volume 31, Issue 2, pp 379–406 | Cite as

Higher order Sobolev homeomorphisms and mappings and the Lusin \({\varvec{(N)}}\) condition

  • Tomáš Roskovec
Article
  • 156 Downloads

Abstract

We give conditions on k, p and the dimension of the space characterizing when the Lusin (N) condition holds in \(W^{k,p}\). We generalize well-known counterexamples in \(W^{1,p}\) both for general mappings and for homeomorphisms.

Keywords

Lusin condition Sobolev space Area formula 

Mathematics Subject Classification

46E35 

Notes

Acknowledgements

I would like to thank Jan Malý who gave me some useful ideas about the problem. And I thank the my supervisor, Stanislav Hencl, for valuable comments. I also thank Thomas Zürcher for help with the final form of the text.

References

  1. 1.
    Bourgain, J., Korobkov, M., Kristensen, J.: On the Morse–Sard property and level sets of Sobolev and BV functions. Rev. Mat. Iberoam. 29(1), 1–23 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bourgain, J., Korobkov, M., Kristensen, J.: On the Morse–Sard property and level sets of \(W^{n,1}\) Sobolev functions on \(R^n\). J. Reine Angew. Math. 700, 93–112 (2015)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Cesari, L.: Sulle transformazioni continue. Ann. Mat. Pura Appl. 21, 157–188 (1942)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    D’Onofrio, L., Hencl, S., Malý, J., Schiattarella, R.: Note on Lusin (N) condition and the distributional determinant. J. Math. Anal. Appl. 439(1), 171–182 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Henao, D., Mora-Corral, C.: Lusin’s condition and the distributional determinant for deformations with finite energy. Adv. Calc. Var. 5, 355–409 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hencl, S.: Sobolev homeomorphism with zero Jacobian almost everywhere. J. Math. Pures Appl. (9) 95(4), 444–458 (2011)Google Scholar
  7. 7.
    Hencl, S., Koskela, P.: Lectures on Mappings of Finite Distortion. Lecture Notes in Mathematics, vol. 2096. Springer, Cham (2014)Google Scholar
  8. 8.
    Kauhanen, J., Koskela, P., Malý, J.: On functions with derivatives in a Lorentz space. Manuscr. Math. 100(1), 87–101 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kauranen, A.: Generalized dimension estimates for images of porous sets in metric spaces. Ann. Acad. Sci. Fenn. Math. 41(1), 85–102 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Korobkov, M., Kristensen, J.: On the Morse–Sard theorem for the sharp case of Sobolev mappings. Indiana Univ. Math. J. 63(6), 1703–1724 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Koskela, P., Malý, J., Zürcher, T.: Luzin’s condition (N) and Sobolev mappings. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 9 Mat. Appl. 23(4), 455–465 (2012)Google Scholar
  12. 12.
    Malý, J., Martio, O.: Lusin’s condition (N) and mappings of the class \(W^{1, n}\). J. Reine Angew. Math. 458, 19–36 (1995)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Marcus, M., Mizel, V.J.: Transformations by functions in Sobolev spaces and lower semicontinuity for parametric variational problems. Bull. Am. Math. Soc. 79, 790–795 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Martio, O., Ziemer, W.P.: Lusin’s condition (N) and mappings with nonnegative Jacobians. Mich. Math. J. 39(3), 495–508 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Matějka, M.: Sobolev mappings and Luzin condition N. Master thesis, Faculty of Mathematics and Physics, Charles University in Prague. https://is.cuni.cz/webapps/zzp/detail/70562/ (2013) (in Czech)
  16. 16.
    Menne, U.: Weakly differentiable functions on varifolds. Indiana Univ. Math. J. 65(3), 977–1088 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Oliva, M.: Bi-Sobolev homeomorphisms \(f\) with \(Df\) and \(Df^{-1}\) of low rank using laminates. Calc. Var. Partial Differ. Equ. 55(6), 55–135 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Peetre, J.: Espaces d’interpolation et théorème de Soboleff. Ann. Inst. Fourier (Grenoble) 16(1), 279–317 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ponomarev, S.: Examples of homeomorphisms in the class \(\mathit{ACTL}^p\) which do not satisfy the absolute continuity condition of Banach. Dokl. Akad. Nauk USSR 201, 1053–1054 (1971). (Russian)Google Scholar
  20. 20.
    Reshetnyak, Y.G.: Certain geometric properties of functions and mappings with generalized derivatives. Sibirsk. Mat. Z. 7, 886–919 (1966)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Rickman, S.: Quasiregular Mappings, Ergebnisse der Mathematik und Ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26. Springer, Berlin (1993)Google Scholar
  22. 22.
    Roskovec, T.: Construction of \(W^{2, n}(\Omega )\) function with gradient violating Lusin (N) condition. Math. Nachr. 289(8–9), 1100–1111 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Universidad Complutense de Madrid 2017

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Informatics, Faculty of EconomicsUniversity of South Bohemia in České BudějoviceČeské BudějoviceCzech Republic

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