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Weak Limit of Homeomorphisms in \(W^{1,n-1}\) and (INV) Condition

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Abstract

Let \(\Omega ,\Omega '\subset {\mathbb {R}}^3\) be Lipschitz domains, let \(f_m:\Omega \rightarrow \Omega '\) be a sequence of homeomorphisms with prescribed Dirichlet boundary condition and \(\sup _m \int _{\Omega }(|Df_m|^2+1/J^2_{f_m})<\infty \). Let f be a weak limit of \(f_m\) in \(W^{1,2}\). We show that f is invertible a.e., and more precisely that it satisfies the (INV) condition of Conti and De Lellis, and thus that it has all of the nice properties of mappings in this class. Generalization to higher dimensions and an example showing sharpness of the condition \(1/J^2_f\in L^1\) are also given. Using this example we also show that, unlike the planar case, the class of weak limits and the class of strong limits of \(W^{1,2}\) Sobolev homeomorphisms in \({\mathbb {R}}^3\) are not the same.

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References

  1. Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000

  2. Ball, J.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal. 63, 337–403, 1977

    Article  MathSciNet  MATH  Google Scholar 

  3. Ball, J.: Global invertibility of Sobolev functions and the interpenetration of matter. Proc. Roy. Soc. Edinb. Sect. A 88(3–4), 315–328, 1981

    Article  MathSciNet  MATH  Google Scholar 

  4. Barchiesi, M., Henao, D., Mora-Corral, C., Rodiac, R.: Harmonic dipoles and the relaxation of the neo-Hookean energy in 3D elasticity. Preprint arXiv:2102.12303

  5. Barchiesi, M., Henao, D., Mora-Corral, C., Rodiac, R.: On the lack of compactness problem in the axisymmetric neo-Hookean model. Arch. Ration. Mech. Anal. 247, 70, 2023. arXiv:2111.07112

  6. Barchiesi, M., Henao, D., Mora-Corral, C.: Local invertibility in Sobolev spaces with applications to nematic elastomers and magnetoelasticity. Arch. Ration. Mech. Anal. 224, 743–816, 2017

    Article  MathSciNet  MATH  Google Scholar 

  7. Brezis, H., Nirenberg, L.: Degree theory and BMO. I. Compact manifolds without boundaries. Sel. Math. 1, 197–263, 1995

    Article  MathSciNet  MATH  Google Scholar 

  8. Bouchala, O., Hencl, S., Molchanova, A.: Injectivity almost everywhere for weak limits of Sobolev homeomorphisms. J. Funct. Anal. 279, 108658, 2020

    Article  MathSciNet  MATH  Google Scholar 

  9. CalderónA, P.: Lebesgue spaces of differentiable functions and distributions. Proceedings of Symposia in Pure Mathematics, Vol. IV, 33–49, 1961

  10. Ciarlet, P.G., Nečas, J.: Injectivity and self-contact in nonlinear elasticity. Arch. Ration. Mech. Anal. 97, 171–188, 1987

    Article  MathSciNet  MATH  Google Scholar 

  11. Conti, S., De Lellis, C.: Some remarks on the theory of elasticity for compressible Neohookean materials. Ann. Sc. Norm. Super. Pisa Cl. Sci. 2, 521–549, 2003

    MathSciNet  MATH  Google Scholar 

  12. Doležalová, A., Hencl, S., Molchanova, A.: Weak limits of homeomorphisms in \(W^{1,n-1}\): invertibility and lower semicontinuity of energy preprint arXiv:2212.06452

  13. De Philippis, G., Pratelli, A.: The closure of planar diffeomorphisms in Sobolev spaces. Ann. Inst. H. Poincare Anal. Non Lineaire 37, 181–224, 2020

    Article  MathSciNet  MATH  ADS  Google Scholar 

  14. Federer, H.: Geometric Measure Theory. Springer, 1969, 1996

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

  16. Giaquinta, M., Modica, G., Souček, J.: Cartesian Currents in the Calculus of Variation. DODELAT

  17. Henao, D., Mora-Corral, C.: Lusin’s condition and the distributional determinant for deformations with finite energy. Adv. Calc. Var. 5, 355–409, 2012

    Article  MathSciNet  MATH  Google Scholar 

  18. Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Clarendon Press, Oxford, 1993

  19. Henao, D., Mora-Corral, C.: Fracture surfaces and the regularity of inverses for BV deformations. Arch. Ration. Mech. Anal. 201, 575–629, 2011

    Article  MathSciNet  MATH  Google Scholar 

  20. Hencl, S., Koskela, P.: Lectures on mappings of finite distortion. Lecture Notes in Mathematics, Vol. 2096. Springer, Berlin, 176, 2014

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Hencl, S., Onninen, J.: Jacobian of weak limits of Sobolev homeomorphisms. Adv. Calc. Var. 11(1), 65–73, 2018

    Article  MathSciNet  MATH  Google Scholar 

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

  24. Iwaniec, T., Onninen, J.: Monotone Sobolev mappings of planar domains and surfaces. Arch. Ration. Mech. Anal. 219, 159–181, 2016

    Article  MathSciNet  MATH  Google Scholar 

  25. Iwaniec, T., Onninen, J.: Limits of Sobolev homeomorphisms. J. Eur. Math. Soc. 19, 473–505, 2017

    Article  MathSciNet  MATH  Google Scholar 

  26. Jones, P.W.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147, 71–88, 1981

    Article  MathSciNet  MATH  Google Scholar 

  27. Koskela, P., Malý, J.: Mappings of finite distortion: the zero set of the Jacobian. J. Eur. Math. Soc. 5, 95–105, 2003

    Article  MathSciNet  MATH  Google Scholar 

  28. Koskela, P., Rajala, T., Zhang, Y.: A geometric characterization of planar Sobolev extension domains. arXiv:1502.04139v6

  29. Koski, A., Onninen, J.: The Sobolev Jordan-Schonflies Problem. Adv. Math. 413, 108795, 2023. arXiv:2008.09947

  30. Krömer, S.: Global invertibility for orientation-preserving Sobolev maps via invertibility on or near the boundary. Arch. Ration. Mech. Anal. 238, 1113–1155, 2020

    Article  MathSciNet  MATH  Google Scholar 

  31. Müller, S., Spector, S.J.: An existence theory for nonlinear elasticity that allows for cavitation. Arch. Ration. Mech. Anal. 131(1), 1–66, 1995

    Article  MathSciNet  MATH  Google Scholar 

  32. Müller, S., Spector, S.J., Tang, Q.: Invertibility and a topological property of Sobolev maps. SIAM J. Math. Anal. 27, 959–976, 1996

    Article  MathSciNet  MATH  Google Scholar 

  33. Müller, S., Tang, Q., Yan, B.S.: On a new class of elastic deformations not allowing for cavitation. Anal. Nonlineaire 11, 217–243, 1994

    MathSciNet  MATH  Google Scholar 

  34. ReshetnyakY, G.: Space Mappings with Bounded Distortion. Transl. Math. Monographs, Vol. 73. AMS, Providence, 1989

  35. SteinE, M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, 1970

  36. Šverák, V.: Regularity properties of deformations with finite energy. Arch. Ration. Mech. Anal. 100(2), 105–127, 1988

    Article  MathSciNet  MATH  Google Scholar 

  37. Swanson, D., Ziemer, W.P.: A topological aspect of Sobolev mappings. Calc. Var. Partial Differ. Equ. 14(1), 69–84, 2002

    Article  MathSciNet  MATH  Google Scholar 

  38. Swanson, D., Ziemer, W.P.: The image of a weakly differentiable mapping. SIAM J. Math. Anal. 35(5), 1099–1109, 2004

    Article  MathSciNet  MATH  Google Scholar 

  39. Tang, Q.: Almost-everywhere injectivity in nonlinear elasticity. Proc. R. Soc. Edinb. Sect. A 109, 79–95, 1988

    Article  MathSciNet  MATH  Google Scholar 

  40. Ziemer, W.P.: Weakly differentiable functions. Graduate texts in Mathematics, Vol. 120. Springer, Berlin, 1989

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Acknowledgements

We would like to thank the referees for carefully reading the manuscript and for many valuable comments which improved its readibility.

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Authors and Affiliations

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

    Anna Doležalová, Stanislav Hencl & Jan Malý

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  1. Anna Doležalová
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  2. Stanislav Hencl
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Correspondence to Stanislav Hencl.

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Communicated by I. Fonseca.

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The authors were supported by the Grant GAČR P201/21-01976S. The first author was also supported in part by the Danube Region Grant No. 8X2043 of the Czech Ministry of Education, Youth and Sports and by the project Grant Schemes at CU, Reg. No. CZ.02.2.69/0.0/0.0/19_073/0016935.

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Doležalová, A., Hencl, S. & Malý, J. Weak Limit of Homeomorphisms in \(W^{1,n-1}\) and (INV) Condition. Arch Rational Mech Anal 247, 80 (2023). https://doi.org/10.1007/s00205-023-01911-7

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