The closure of planar diffeomorphisms in Sobolev spaces
Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 1, pp. 181-224.
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We characterize the (sequentially) weak and strong closure of planar diffeomorphisms in the Sobolev topology and we show that they always coincide. We also provide some sufficient condition for a planar map to be approximable by diffeomorphisms in terms of the connectedness of its counter-images, in the spirit of Young's characterisation of monotone functions. We finally show that the closure of diffeomorphisms in the Sobolev topology is strictly contained in the class INV introduced by Müller and Spector.

DOI : 10.1016/j.anihpc.2019.08.001
Mots-clés : Sobolev approximation of mappings, INV mappings, Non-crossing mappings 
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     title = {The closure of planar diffeomorphisms in {Sobolev} spaces},
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De Philippis, G.; Pratelli, A. The closure of planar diffeomorphisms in Sobolev spaces. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 1, pp. 181-224. doi : 10.1016/j.anihpc.2019.08.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2019.08.001/

[1] Alessandrini, G.; Sigalotti, M. Geometric properties of solutions to the anisotropic p -Laplace equation in dimension two, Ann. Acad. Sci. Fenn., Math., Volume 26 (2001) no. 1, pp. 249–266 | MR | Zbl

[2] Ball, J.M. Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. R. Soc. Lond. A, Volume 306 (1982), pp. 557–611 | MR | Zbl

[3] Barchiesi, M.; Henao, D.; Mora-Corral, C. Local invertibility in Sobolev spaces with applications to nematic elastomers and magnetoelasticity, Arch. Ration. Mech. Anal., Volume 224 (2017) no. 2, pp. 743–816 | DOI | MR | Zbl

[4] Campbell, D. Diffeomorphic approximation of planar Sobolev homeomorphisms in Orlicz-Sobolev spaces, J. Funct. Anal., Volume 273 (2017) no. 1, pp. 125–205 | DOI | MR | Zbl

[5] D. Campbell, J. Onninen, A. Rabina, V. Tengvall, Diffeomorphic approximation of W1,1 planar Sobolev monotone maps, 2017, in preparation.

[6] Conti, S.; De Lellis, C. Some remarks on the theory of elasticity for compressible Neohookean materials, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 2 (2003), pp. 521–549 | Numdam | MR | Zbl

[7] Daneri, S.; Pratelli, A. Smooth approximation of bi-Lipschitz orientation-preserving homeomorphisms, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 31 (2014), pp. 567–589 | DOI | Numdam | MR | Zbl

[8] Evans, L.C.; Gariepy, R.F. Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992 | MR | Zbl

[9] Giaquinta, M.; Modica, G.; Souček, J. Cartesian Currents in Calculus of Variations I and II, Springer, 1998 | MR | Zbl

[10] Henao, D. Cavitation, invertibility, and convergence of regularized minimizers in nonlinear elasticity, J. Elast., Volume 94 (2009) no. 1, pp. 55–68 | DOI | MR | Zbl

[11] Hencl, S.; Pratelli, A. Diffeomorphic approximation of W1,1 planar Sobolev homeomorphisms, J. Eur. Math. Soc., Volume 20 (2018) no. 3, pp. 597–656 | DOI | MR | Zbl

[12] Iwaniec, T.; Kovalev, L.; Onninen, J. Diffeomorphic approximation of Sobolev homeomorphisms, Arch. Ration. Mech. Anal., Volume 201 (2011) no. 3, pp. 1047–1067 | DOI | MR | Zbl

[13] Iwaniec, T.; Kovalev, L.; Onninen, J. Hopf differentials and smoothing Sobolev homeomorphisms, Int. Math. Res. Not., Volume 14 (2012), pp. 3256–3277 | MR | Zbl

[14] Iwaniec, T.; Onninen, J. Monotone Sobolev mappings of planar domains and surfaces, Arch. Ration. Mech. Anal., Volume 219 (2016) no. 1, pp. 159–181 | DOI | MR | Zbl

[15] Iwaniec, T.; Onninen, J. Limits of Sobolev homeomorphisms, J. Eur. Math. Soc., Volume 19 (2017) no. 2, pp. 473–505 | DOI | MR | Zbl

[16] Mora Corral, C.; Pratelli, A. Approximation of piecewise affine homeomorphisms by diffeomorphisms, J. Geom. Anal., Volume 24 (2014) no. 3, pp. 1398–1424 | DOI | MR | Zbl

[17] Müller, S.; Spector, S.J. An existence theory for nonlinear elasticity that allows for cavitation, Arch. Ration. Mech. Anal., Volume 131 (1995), pp. 1–66 | DOI | MR | Zbl

[18] Radici, E. A planar Sobolev extension theorem for piecewise linear homeomorphisms, Pac. J. Math., Volume 283 (2016) no. 2, pp. 405–418 | DOI | MR | Zbl

[19] Šverák, V. Regularity properties of deformations with finite energy, Arch. Ration. Mech. Anal., Volume 100 (1988), pp. 105–127 | DOI | MR | Zbl

[20] Youngs, J.W.T. Homeomorphic approximations to monotone mappings, Duke Math. J., Volume 15 (1948), pp. 87–94 | MR | Zbl

[21] Youngs, J.W.T. The topological theory of Fréchet surfaces, Ann. Math., Volume 45 (1944), pp. 753–785 | MR | Zbl

[22] Youngs, J.W.T. A reduction theorem concerning the representation problem for Fréchet varieties, Proc. Natl. Acad. Sci. USA, Volume 32 (1946), pp. 328–330 | MR | Zbl

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