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

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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 J f  < 0 on a set of positive measure. It follows that there are no diffeomorphisms (or piecewise affine homeomorphisms) f k such that \({f_k\to f}\) in \({W^{1,1}_{\rm loc}}\).

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References

  1. Acerbi E., Fusco N.: Semicontinuity problems in the calculus of variations. Arch. Rat. Mech. Anal. 86, 125–145 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  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, 2000

  3. Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. In: Arch. Rat. Mech. Anal. 63, 337–403 (1977)

  4. Ball J.M.: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos. Trans. R. Soc. Lond. A 306(1496), 557–611 (1982)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  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, 2001

  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, 2010

  7. Bellido J.C., Mora-Corral C.: Approximation of Hölder continuous homeomorphisms by piecewise affine homeomorphisms. Houston J. Math. 37(2), 449–500 (2011)

    MATH  MathSciNet  Google Scholar 

  8. Bing R.H.: Locally tame sets are tame. Ann. Math. 59, 145–158 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  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. Černý R.: Homeomorphism with zero Jacobian: Sharp integrability of the derivative. J. Math. Anal. Appl 373, 161–174 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Connell E.H.: Approximating stable homeomorphisms by piecewise linear ones. Ann. Math. 78, 326–338 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  13. Daneri, S., Pratelli, A.: A planar bi-Lipschitz extension theorem. Adv. Calc. Var. 2014 (to appear)

  14. Donaldson S.K., Sullivan D.P.: Quasiconformal 4-manifolds. Acta Math. 163, 181–252 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  15. D’Onofrio L., Hencl S., Schiattarella R.: Bi-Sobolev homeomorphism with zero Jacobian almost everywhere. Calc. Var. PDE 51, 139–170 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  16. Evans L.C.: Quasiconvexity and partial regularity in the calculus of variations. Ann. Math. 95(3), 227–252 (1986)

    MATH  MathSciNet  Google Scholar 

  17. Fonseca, I., Gangbo, W.: Degree Theory in Analysis and Applications. Clarendon Press, Oxford, 1995

  18. Goldstein, P., Hajlasz, P.: An Orientation-Preserving Homeomorphism of a Cube may have a.e. Negative Jacobian, 2013 (preprint)

  19. Hencl S.: Sobolev homeomorphism with zero Jacobian almost everywhere. J. Math. Pures Appl. 95, 444–458 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hencl, S., Koskela, P.: Lectures on mappings of finite distortion. In: Lecture Notes in Mathematics, vol. 2096. Springer, Berlin, 2014

  21. Hencl S., Malý J.: Jacobians of Sobolev homeomorphisms. Calc. Var. Par. Differ. Equ. 38, 233–242 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hencl, S., Pratelli, A.: Diffeomorphic Approximation of W 1,1 Planar Sobolev Homeomorphism., 2015 (preprint)

  23. Iwaniec, T., Martin, G.: Geometric function theory and nonlinear analysis. In: Oxford Mathematical Monographs. Clarendon Press, Oxford, 2001

  24. Iwaniec, T., Onninen, J.: Limits of Sobolev homeomorphisms. J. Eur. Math. Soc. 2014 (preprint)

  25. Iwaniec T., Kovalev L.V., Onninen J.: Diffeomorphic approximation of Sobolev homeomorphisms. Arch. Rat. Mech. Anal. 201(3), 1047–1067 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  26. Iwaniec T., Kovalev L.V., Onninen J.: Hopf Differentials and smoothing Sobolev homeomorphisms. Int. Math. Res. Not. 14, 3256–3277 (2012)

    MathSciNet  MATH  Google Scholar 

  27. Iwaniec, T., Onninen, J.: Monotone Sobolev Mappings of Planar Domains and Surfaces, 2014 (preprint)

  28. Kirby R.C.: Stable homeomorphisms and the annulus conjecture. Ann. Math. 89, 575–582 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  30. Koskela, P.: Lectures on Quasiconformal Mappings. University of Jyväskylä, Jyväskylä

  31. Luukkainen J.: Lipschitz and quasiconformal approximation of homeomorphism pairs. Topol. Appl. 109, 1–40 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  32. Moise E.E.: Affine structures in 3-manifolds. IV. Piecewise linear approximations of homeomorphisms. Ann. Math. 55, 215–222 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  33. Moise, E.E.: Geometric topology in dimensions 2 and 3. Graduate Texts in Mathematics, vol. 47. Springer, New York, 1977

  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)

    MATH  MathSciNet  Google Scholar 

  35. Morrey C.B.: Quasi-convexity and the semicontinuity of multiple integrals. Pac. J. Math. 2, 25–53 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  37. Radó T.: Über den Begriff Riemannschen Fläche. Acta. Math. Szeged 2, 101–121 (1925)

    MATH  Google Scholar 

  38. Rado, T., Reichelderfer, P.V.: Continuous Transformations in Analysis. Springer, Berlin, 1955

  39. Rushing, T.B.: Topological embeddings. Pure and Applied Mathematics, vol. 52. Academic Press, New York, 1973

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Authors and Affiliations

  1. Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 00, Prague 8, Czech Republic

    Stanislav Hencl & Benjamin Vejnar

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  1. Stanislav Hencl
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  2. Benjamin Vejnar
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Correspondence to Stanislav Hencl.

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Communicated by V. Šverák

S. Hencl was supported by the ERC CZ Grant LL1203 of the Czech Ministry of Education. B. Vejnar was supported by the Grant GAČR 14-06989P and he is a junior researcher in the University Center for Mathematical Modeling, Applied Analysis and Computational Mathematics (Math MAC).

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Hencl, S., Vejnar, B. Sobolev Homeomorphism that Cannot be Approximated by Diffeomorphisms in W1,1 . Arch Rational Mech Anal 219, 183–202 (2016). https://doi.org/10.1007/s00205-015-0895-5

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  • DOI: https://doi.org/10.1007/s00205-015-0895-5

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