Archive for Rational Mechanics and Analysis

, Volume 219, Issue 1, pp 183–202 | Cite as

Sobolev Homeomorphism that Cannot be Approximated by Diffeomorphisms in W1,1

Article

Abstract

We construct a Sobolev homeomorphism in dimension \({n \geqq 4,\,f \in W^{1,1}((0, 1)^n,\mathbb{R}^n)}\) such that \({J_f = {\rm det} Df > 0}\) on a set of positive measure and Jf < 0 on a set of positive measure. It follows that there are no diffeomorphisms (or piecewise affine homeomorphisms) fk such that \({f_k\to f}\) in \({W^{1,1}_{\rm loc}}\).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Acerbi E., Fusco N.: Semicontinuity problems in the calculus of variations. Arch. Rat. Mech. Anal. 86, 125–145 (1984)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. In: Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000Google Scholar
  3. 3.
    Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. In: Arch. Rat. Mech. Anal. 63, 337–403 (1977)Google Scholar
  4. 4.
    Ball J.M.: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos. Trans. R. Soc. Lond. A 306(1496), 557–611 (1982)MATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Ball, J.M.: Singularities and computation of minimizers for variational problems. In: Foundations of Computational Mathematics, Oxford, 1999, pp. 1–20, London Math. Soc. Lecture Note Ser., vol. 284. Cambridge Univ. Press, Cambridge, 2001Google Scholar
  6. 6.
    Ball, J.M.: Progress and puzzles in nonlinear elasticity. Proceedings of course on Poly-, Quasi- and Rank-One Convexity in Applied Mechanics. CISM Courses and Lectures. Springer, Berlin, 2010Google Scholar
  7. 7.
    Bellido J.C., Mora-Corral C.: Approximation of Hölder continuous homeomorphisms by piecewise affine homeomorphisms. Houston J. Math. 37(2), 449–500 (2011)MATHMathSciNetGoogle Scholar
  8. 8.
    Bing R.H.: Locally tame sets are tame. Ann. Math. 59, 145–158 (1954)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Bing R.H.: Stable homeomorphisms on E 5 can be approximated by piecewise linear ones. Not. Am. Math. Soc. 10, 607–616 (1963)Google Scholar
  10. 10.
    Černý R.: Homeomorphism with zero Jacobian: Sharp integrability of the derivative. J. Math. Anal. Appl 373, 161–174 (2011)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Connell E.H.: Approximating stable homeomorphisms by piecewise linear ones. Ann. Math. 78, 326–338 (1963)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Daneri, S., Pratelli, A.: Smooth approximation of bi-Lipschitz orientation-preserving homeomorphisms. Ann. Inst. H. Poincaré Anal. Non Linéaire 31(3), 567–589 (2014)Google Scholar
  13. 13.
    Daneri, S., Pratelli, A.: A planar bi-Lipschitz extension theorem. Adv. Calc. Var. 2014 (to appear)Google Scholar
  14. 14.
    Donaldson S.K., Sullivan D.P.: Quasiconformal 4-manifolds. Acta Math. 163, 181–252 (1989)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    D’Onofrio L., Hencl S., Schiattarella R.: Bi-Sobolev homeomorphism with zero Jacobian almost everywhere. Calc. Var. PDE 51, 139–170 (2014)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Evans L.C.: Quasiconvexity and partial regularity in the calculus of variations. Ann. Math. 95(3), 227–252 (1986)MATHMathSciNetGoogle Scholar
  17. 17.
    Fonseca, I., Gangbo, W.: Degree Theory in Analysis and Applications. Clarendon Press, Oxford, 1995Google Scholar
  18. 18.
    Goldstein, P., Hajlasz, P.: An Orientation-Preserving Homeomorphism of a Cube may have a.e. Negative Jacobian, 2013 (preprint)Google Scholar
  19. 19.
    Hencl S.: Sobolev homeomorphism with zero Jacobian almost everywhere. J. Math. Pures Appl. 95, 444–458 (2011)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Hencl, S., Koskela, P.: Lectures on mappings of finite distortion. In: Lecture Notes in Mathematics, vol. 2096. Springer, Berlin, 2014Google Scholar
  21. 21.
    Hencl S., Malý J.: Jacobians of Sobolev homeomorphisms. Calc. Var. Par. Differ. Equ. 38, 233–242 (2010)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Hencl, S., Pratelli, A.: Diffeomorphic Approximation of W 1,1 Planar Sobolev Homeomorphism., 2015 (preprint)Google Scholar
  23. 23.
    Iwaniec, T., Martin, G.: Geometric function theory and nonlinear analysis. In: Oxford Mathematical Monographs. Clarendon Press, Oxford, 2001Google Scholar
  24. 24.
    Iwaniec, T., Onninen, J.: Limits of Sobolev homeomorphisms. J. Eur. Math. Soc. 2014 (preprint)Google Scholar
  25. 25.
    Iwaniec T., Kovalev L.V., Onninen J.: Diffeomorphic approximation of Sobolev homeomorphisms. Arch. Rat. Mech. Anal. 201(3), 1047–1067 (2011)MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Iwaniec T., Kovalev L.V., Onninen J.: Hopf Differentials and smoothing Sobolev homeomorphisms. Int. Math. Res. Not. 14, 3256–3277 (2012)MathSciNetMATHGoogle Scholar
  27. 27.
    Iwaniec, T., Onninen, J.: Monotone Sobolev Mappings of Planar Domains and Surfaces, 2014 (preprint)Google Scholar
  28. 28.
    Kirby R.C.: Stable homeomorphisms and the annulus conjecture. Ann. Math. 89, 575–582 (1969)MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Kirby, R.C., Siebenmann, L.C., Wall, C.T.C.: The annulus conjecture and triangulation. Not. Am. Math. Soc. 16(432), abstract 69T-G27 (1969)Google Scholar
  30. 30.
    Koskela, P.: Lectures on Quasiconformal Mappings. University of Jyväskylä, JyväskyläGoogle Scholar
  31. 31.
    Luukkainen J.: Lipschitz and quasiconformal approximation of homeomorphism pairs. Topol. Appl. 109, 1–40 (2001)MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Moise E.E.: Affine structures in 3-manifolds. IV. Piecewise linear approximations of homeomorphisms. Ann. Math. 55, 215–222 (1952)MATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Moise, E.E.: Geometric topology in dimensions 2 and 3. Graduate Texts in Mathematics, vol. 47. Springer, New York, 1977Google Scholar
  34. 34.
    Mora-Corral C.: Approximation by piecewise affine homeomorphisms of Sobolev homeomorphisms that are smooth outside a point. Houston J. Math. 35, 515–539 (2009)MATHMathSciNetGoogle Scholar
  35. 35.
    Morrey C.B.: Quasi-convexity and the semicontinuity of multiple integrals. Pac. J. Math. 2, 25–53 (1952)MATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Müller S., Tang Q., Yan B.S.: On a new class of elastic deformations not allowing for cavitation. Ann. Inst. Henri Poincaré 11, 217–243 (1994)MATHMathSciNetGoogle Scholar
  37. 37.
    Radó T.: Über den Begriff Riemannschen Fläche. Acta. Math. Szeged 2, 101–121 (1925)MATHGoogle Scholar
  38. 38.
    Rado, T., Reichelderfer, P.V.: Continuous Transformations in Analysis. Springer, Berlin, 1955Google Scholar
  39. 39.
    Rushing, T.B.: Topological embeddings. Pure and Applied Mathematics, vol. 52. Academic Press, New York, 1973Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematical AnalysisCharles UniversityPrague 8Czech Republic

Personalised recommendations