Smooth approximation of bi-Lipschitz orientation-preserving homeomorphisms

Daneri, Sara; Pratelli, Aldo

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 3, pp. 567-589

We show that a planar bi-Lipschitz orientation-preserving homeomorphism can be approximated in the $W^{1,p}$ norm, together with its inverse, with an orientation-preserving homeomorphism which is piecewise affine or smooth.

[1] E. Acerbi, N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Ration. Mech. Anal. 86 (1984), 125 -145 | MR | Zbl

[2] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal. 63 (1977), 337 -403 | MR | Zbl

[3] J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. R. Soc. Lond. A 306 no. 1496 (1982), 557 -611 | MR | Zbl

[4] J.M. Ball, Singularities and computation of minimizers for variational problems, Foundations of Computational Mathematics, Oxford, 1999, London Math. Soc. Lecture Note Ser. vol. 284 , Cambridge Univ. Press, Cambridge (2001), 1 -20 | Zbl

[5] J.M. Ball, Progress and puzzles in nonlinear elasticity, Proceedings of Course on Poly-, Quasi- and Rank-One Convexity in Applied Mechanics, CISM, Udine (2010), http://dx.doi.org/10.1007/978-3-7091-0174-2

[6] J.C. Bellido, C. Mora-Corral, Approximation of Hölder continuous homeomorphisms by piecewise affine homeomorphisms, Houston J. Math. 37 no. 2 (2011), 449 -500 | MR | Zbl

[7] R.H. Bing, Locally tame sets are tame, Ann. of Math. 59 (1954), 145 -158 | MR | Zbl

[8] R.H. Bing, Stable homeomorphisms on $E^5$ can be approximated by piecewise linear ones, Notices Amer. Math. Soc. 10 (1963), 666

[9] E.H. Connell, Approximating stable homeomorphisms by piecewise linear ones, Ann. of Math. 78 (1963), 326 -338 | MR | Zbl

[10] M. Csornyei, S. Hencl, J. Maly, Homeomorphisms in the Sobolev space $W^{1,n-1}$ , J. Reine Angew. Math. 644 (2010), 221 -235 | MR | Zbl

[11] S. Daneri, A. Pratelli, A planar bi-Lipschitz extension theorem, Adv. Calc. Var. (2013) | MR | Zbl

[12] P. Di Gironimo, L. D'Onofrio, C. Sbordone, R. Schiattarella, Anisotropic Sobolev homeomorphisms, Ann. Acad. Sci. Fenn. Math. 36 no. 2 (2011), 593 -602 | MR | Zbl

[13] S.K. Donaldson, D.P. Sullivan, Quasiconformal 4-manifolds, Acta Math. 163 (1989), 181 -252 | MR | Zbl

[14] L.C. Evans, Quasiconvexity and partial regularity in the calculus of variations, Arch. Ration. Mech. Anal. 95 no. 3 (1986), 227 -252 | MR | Zbl

[15] S. Hencl, Sharpness of the assumptions for the regularity of a homeomorphism, Michigan Math. J. 59 no. 3 (2010), 667 -678 | MR | Zbl

[16] S. Hencl, Sobolev homeomorphism with zero Jacobian almost everywhere, J. Math. Pures Appl. 95 (2011), 444 -458 | MR | Zbl

[17] S. Hencl, P. Koskela, Regularity of the inverse of a planar Sobolev homeomorphism, Arch. Ration. Mech. Anal. 180 (2006), 75 -95 | MR | Zbl

[18] S. Hencl, P. Koskela, J. Maly, Regularity of the inverse of a Sobolev homeomorphism in space, Proc. Roy. Soc. Edinburgh Sect. A 136A no. 6 (2006), 1267 -1285 | MR | Zbl

[19] S. Hencl, J. Maly, Jacobians of Sobolev homeomorphisms, Calc. Var. Partial Differential Equations 38 (2010), 233 -242 | MR | Zbl

[20] S. Hencl, G. Moscariello, A. Passarelli Di Napoli, C. Sbordone, Bi-Sobolev mappings and elliptic equations in the plane, J. Math. Anal. Appl. 355 (2009), 22 -32 | MR | Zbl

[21] T. Iwaniec, L.V. Kovalev, J. Onninen, Hopf differentials and smoothing Sobolev homeomorphisms, Int. Math. Res. Not. IMRN 2012 no. 14 (2012), 3256 -3277 | MR | Zbl

[22] T. Iwaniec, L.V. Kovalev, J. Onninen, Diffeomorphic approximation of Sobolev homeomorphisms, Arch. Ration. Mech. Anal. 201 no. 3 (2011), 1047 -1067 | MR | Zbl

[23] R.C. Kirby, Stable homeomorphisms and the annulus conjecture, Ann. of Math. 89 (1969), 575 -582 | MR | Zbl

[24] R.C. Kirby, L.C. Siebenmann, C.T.C. Wall, The annulus conjecture and triangulation, Notices Amer. Math. Soc. 16 (1969), 432

[25] J. Luukkainen, Lipschitz and quasiconformal approximation of homeomorphism pairs, Topology Appl. 109 (2001), 1 -40 | MR | Zbl

[26] E.E. Moise, Affine structures in 3-manifolds. IV. Piecewise linear approximations of homeomorphisms, Ann. of Math. 55 (1952), 215 -222 | MR | Zbl

[27] E.E. Moise, Geometric Topology in Dimensions 2 and 3, Grad. Texts in Math. vol. 47 , Springer, New York, Heidelberg (1977) | MR | Zbl

[28] C. Mora-Corral, Approximation by piecewise affine homeomorphisms of Sobolev homeomorphisms that are smooth outside a point, Houston J. Math. 35 no. 2 (2009), 515 -539 | MR | Zbl

[29] C. Mora-Corral, A. Pratelli, Approximation of piecewise affine homeomorphisms by diffeomorphisms, J. Geom. Anal. (2013), http://dx.doi.org/10.1007/s12220-012-9378-1 | MR | Zbl

[30] C.B. Morrey, Quasi-convexity and the semicontinuity of multiple integrals, Pacific J. Math. 2 (1952), 25 -53 | MR | Zbl

[31] S. Müller, T. Qi, B.S. Yan, On a new class of elastic deformations not allowing for cavitation, Ann. Inst. Henri Poincaré 11 (1994), 217 -243 | MR | EuDML | Zbl | Numdam

[32] T. Radó, Über den Begriff Riemannschen Fläche, Acta. Math. Szeged 2 (1925), 101 -121 | JFM

[33] T.B. Rushing, Topological Embeddings, Pure Appl. Math. vol. 52 , Academic Press, New York, London (1973) | MR | Zbl

[34] G.A. Seregin, T.N. Shilkin, Some remarks on the mollification of piecewise-linear homeomorphisms, J. Math. Sci. (New York) 87 (1997), 3428 -3433 | MR | Zbl

[35] P. Tukia, The planar Schönflies theorem for Lipschitz maps, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 no. 1 (1980), 49 -72 | MR | Zbl