Invertibility of Sobolev mappings under minimal hypotheses

Kovalev, Leonid V.; Onninen, Jani; Rajala, Kai

doi:10.1016/j.anihpc.2009.09.010

Kovalev, Leonid V. ; Onninen, Jani ; Rajala, Kai

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 2, pp. 517-528.

Résumé

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.

MR Zbl

DOI : 10.1016/j.anihpc.2009.09.010

Classification : 30C65, 26B10, 26B25
Mots clés : Local homeomorphism, Differential inclusion, Finite distortion

@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},
     pages = {517--528},
     publisher = {Elsevier},
     volume = {27},
     number = {2},
     year = {2010},
     doi = {10.1016/j.anihpc.2009.09.010},
     mrnumber = {2595190},
     zbl = {1190.30019},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2009.09.010/}
}

TY  - JOUR
AU  - Kovalev, Leonid V.
AU  - Onninen, Jani
AU  - Rajala, Kai
TI  - Invertibility of Sobolev mappings under minimal hypotheses
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2010
SP  - 517
EP  - 528
VL  - 27
IS  - 2
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2009.09.010/
DO  - 10.1016/j.anihpc.2009.09.010
LA  - en
ID  - AIHPC_2010__27_2_517_0
ER  -

%0 Journal Article
%A Kovalev, Leonid V.
%A Onninen, Jani
%A Rajala, Kai
%T Invertibility of Sobolev mappings under minimal hypotheses
%J Annales de l'I.H.P. Analyse non linéaire
%D 2010
%P 517-528
%V 27
%N 2
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2009.09.010/
%R 10.1016/j.anihpc.2009.09.010
%G en
%F 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) no. 2, pp. 517-528. doi : 10.1016/j.anihpc.2009.09.010. http://archive.numdam.org/articles/10.1016/j.anihpc.2009.09.010/

Bibliographie
Cité par

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

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

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

[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 | EuDML | Numdam | MR | Zbl

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

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

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

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

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

[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 | Zbl

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

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

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

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

[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 | Zbl

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

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

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

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

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

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

Cité par Sources :