Invertibility of Sobolev mappings under minimal hypotheses
Kovalev, Leonid V. ; Onninen, Jani ; Rajala, Kai
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010), p. 517-528 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

We prove a version of the Inverse Function Theorem for continuous weakly differentiable mappings. Namely, a nonconstant W 1,n mapping is a local homeomorphism if it has integrable inner distortion function and satisfies a certain differential inclusion. The integrability assumption is shown to be optimal.

Publié le : 2010-01-01
DOI : https://doi.org/10.1016/j.anihpc.2009.09.010
Classification:  30C65,  26B10,  26B25 
@article{AIHPC_2010__27_2_517_0,
     author = {Kovalev, Leonid V. and Onninen, Jani and Rajala, Kai},
     title = {Invertibility of Sobolev mappings under minimal hypotheses},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {27},
     year = {2010},
     pages = {517-528},
     doi = {10.1016/j.anihpc.2009.09.010},
     mrnumber = {2595190},
     zbl = {1190.30019},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2010__27_2_517_0}
}

Kovalev, Leonid V.; Onninen, Jani; Rajala, Kai. Invertibility of Sobolev mappings under minimal hypotheses. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) pp. 517-528. doi : 10.1016/j.anihpc.2009.09.010. http://gdmltest.u-ga.fr/item/AIHPC_2010__27_2_517_0/

[1] K. Astala, T. Iwaniec, G.J. Martin, J. Onninen, Extremal mappings of finite distortion, Proc. London Math. Soc. (3) 91 no. 3 (2005), 655-702 | MR 2180459 | Zbl 1089.30013

[2] J.M. Ball, Global invertibility of Sobolev functions and the interpenetration of matter, Proc. Roy. Soc. Edinburgh Sect. A 88 no. 3–4 (1981), 315-328 | MR 616782 | Zbl 0478.46032

[3] V.M. Gol'Dshtein, The behavior of mappings with bounded distortion when the distortion coefficient is close to one, Sibirsk. Mat. Zh. 12 (1971), 1250-1258 | MR 294634 | Zbl 0231.30031

[4] V.M. Gol'Dshtein, S.K. Vodopyanov, Quasiconformal mappings, and spaces of functions with first generalized derivatives, Sibirsk. Mat. Zh. 17 no. 3 (1976), 515-531 | MR 414869

[5] J. Heinonen, T. Kilpeläinen, BLD-mappings in W 2,2 are locally invertible, Math. Ann. 318 no. 2 (2000), 391-396 | MR 1795568 | Zbl 0967.30015

[6] J. Heinonen, P. Koskela, Sobolev mappings with integrable dilatations, Arch. Ration. Mech. Anal. 125 no. 1 (1993), 81-97 | MR 1241287 | Zbl 0792.30016

[7] S. Hencl, P. Koskela, Mappings of finite distortion: Discreteness and openness for quasi-light mappings, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 no. 3 (2005), 331-342 | Numdam | MR 2136727 | Zbl 1076.30024

[8] S. Hencl, J. Malý, Mappings of finite distortion: Hausdorff measure of zero sets, Math. Ann. 324 no. 3 (2002), 451-464 | MR 1938454 | Zbl 1017.30030

[9] T. Iwaniec, G. Martin, Geometric Function Theory and Non-Linear Analysis, Oxford Univ. Press, New York (2001) | MR 1859913

[10] T. Iwaniec, V. Šverák, On mappings with integrable dilatation, Proc. Amer. Math. Soc. 118 no. 1 (1993), 181-188 | MR 1160301 | Zbl 0784.30015

[11] F. John, On quasi-isometric mappings. I, Comm. Pure Appl. Math. 21 (1968), 77-110 | MR 222666 | Zbl 0157.45803

[12] P. Koskela, J. Onninen, Mappings of finite distortion: Capacity and modulus inequalities, J. Reine Angew. Math. 599 (2006), 1-26 | MR 2279096 | Zbl 1114.30021

[13] P. Koskela, J. Onninen, K. Rajala, Mappings of finite distortion: Injectivity radius of a local homeomorphism, Future Trends in Geometric Function Theory, Rep. Univ. Jyväskylä Dep. Math. Stat. vol. 92, Univ. Jyväskylä, Jyväskylä (2003), 169-174 | MR 2061002 | Zbl 1043.30009

[14] L.V. Kovalev, J. Onninen, On invertibility of Sobolev mappings, preprint, 2008, arXiv:0812.2350 | MR 2818853

[15] J.J. Manfredi, E. Villamor, An extension of Reshetnyak's theorem, Indiana Univ. Math. J. 47 no. 3 (1998), 1131-1145 | MR 1665761 | Zbl 0931.30014

[16] O. Martio, S. Rickman, J. Väisälä, Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I 488 (1971) | MR 299782 | Zbl 0223.30018

[17] J. Onninen, Mappings of finite distortion: Minors of the differential matrix, Calc. Var. Partial Differential Equations 21 no. 4 (2004), 335-348 | MR 2098071 | Zbl 1072.30016

[18] K. Rajala, The local homeomorphism property of spatial quasiregular mappings with distortion close to one, Geom. Funct. Anal. 15 no. 5 (2005), 1100-1127 | MR 2221159 | Zbl 1096.30019

[19] K. Rajala, Reshetnyak's theorem and the inner distortion, Pure Appl. Math. Q., in press, University of Jyväskylä preprint, No. 336, 2007 | MR 2815379

[20] K. Rajala, Remarks on the Iwaniec–Šverák conjecture, University of Jyväskylä preprint, No. 377, 2009 | MR 2919747

[21] S. Rickman, Quasiregular Mappings, Springer-Verlag, Berlin (1993) | MR 1238941 | Zbl 0796.30018

[22] Q. Tang, Almost-everywhere injectivity in nonlinear elasticity, Proc. Roy. Soc. Edinburgh Sect. A 109 no. 1–2 (1988), 79-95 | MR 952330 | Zbl 0656.73010

[23] J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Lecture Notes in Math. vol. 229, Springer-Verlag, Berlin (1971) | MR 454009 | Zbl 0221.30031

[24] V.A. Zorich, M.A. Lavrentyev's theorem on quasiconformal space maps, Mat. Sb. (N.S.) 74 no. 116 (1967), 417-433 | MR 223569 | Zbl 0181.08701