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Invertibility of Sobolev mappings under minimal hypotheses. (English) Zbl 1190.30019

The paper deals with a version of the inverse function theorem for continuous weakly differentiable mappings \(f : \Omega \rightarrow {\mathbb R}^n\) with \(\Omega\) being a bounded domain in \({\mathbb R}^n\).
The main result of the paper is the following
Theorem. Suppose \(f\in W_{\text{loc}}^{1, n}(\Omega, {\mathbb R}^n\) is a nonconstant mapping such that the mean inner distortion \({\mathcal K}_{\Omega} [f] < \infty\). If there exists \(\delta > - 1\) such that \(D f\in {\mathcal M}(\delta)\) for almost every \(x\in \Omega\), then \(f\) is a local homeomorphism.
The proof of this theorem is based on two results of independent interest. The first step consists in the proof that the mapping is discrete and open; that is, preimages of points are discrete sets, and images of open sets are open.
Another ingredient of the proof is an estimate for the multiplicity \(N(y, f, A)\) \(:=\) \( \sharp \left(f^{-1}(y)\cap A\right)\) of a local homeomorphism \(f\) in terms of the integral of the inner distortion \({\mathcal K}_{I}(\cdot, f)\) in dimension \(n\geq 3\).

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
26B10 Implicit function theorems, Jacobians, transformations with several variables
26B25 Convexity of real functions of several variables, generalizations
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[1] Astala, K.; Iwaniec, T.; Martin, G. J.; Onninen, J., Extremal mappings of finite distortion, Proc. London Math. Soc. (3), 91, 3, 655-702 (2005) · Zbl 1089.30013
[2] Ball, J. M., Global invertibility of Sobolev functions and the interpenetration of matter, Proc. Roy. Soc. Edinburgh Sect. A, 88, 3-4, 315-328 (1981) · Zbl 0478.46032
[3] Gol’dshtein, V. M., The behavior of mappings with bounded distortion when the distortion coefficient is close to one, Sibirsk. Mat. Zh., 12, 1250-1258 (1971) · Zbl 0231.30031
[4] Gol’dshtein, V. M.; Vodopyanov, S. K., Quasiconformal mappings, and spaces of functions with first generalized derivatives, Sibirsk. Mat. Zh., 17, 3, 515-531 (1976)
[5] Heinonen, J.; Kilpeläinen, T., BLD-mappings in \(W^{2, 2}\) are locally invertible, Math. Ann., 318, 2, 391-396 (2000) · Zbl 0967.30015
[6] Heinonen, J.; Koskela, P., Sobolev mappings with integrable dilatations, Arch. Ration. Mech. Anal., 125, 1, 81-97 (1993) · Zbl 0792.30016
[7] Hencl, S.; Koskela, P., Mappings of finite distortion: Discreteness and openness for quasi-light mappings, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22, 3, 331-342 (2005) · Zbl 1076.30024
[8] Hencl, S.; Malý, J., Mappings of finite distortion: Hausdorff measure of zero sets, Math. Ann., 324, 3, 451-464 (2002) · Zbl 1017.30030
[9] Iwaniec, T.; Martin, G., Geometric Function Theory and Non-Linear Analysis (2001), Oxford Univ. Press: Oxford Univ. Press New York
[10] Iwaniec, T.; Šverák, V., On mappings with integrable dilatation, Proc. Amer. Math. Soc., 118, 1, 181-188 (1993) · Zbl 0784.30015
[11] John, F., On quasi-isometric mappings. I, Comm. Pure Appl. Math., 21, 77-110 (1968) · Zbl 0157.45803
[12] Koskela, P.; Onninen, J., Mappings of finite distortion: Capacity and modulus inequalities, J. Reine Angew. Math., 599, 1-26 (2006) · Zbl 1114.30021
[13] Koskela, P.; Onninen, J.; Rajala, K., Mappings of finite distortion: Injectivity radius of a local homeomorphism, (Future Trends in Geometric Function Theory. Future Trends in Geometric Function Theory, Rep. Univ. Jyväskylä Dep. Math. Stat., vol. 92 (2003), Univ. Jyväskylä: Univ. Jyväskylä Jyväskylä), 169-174 · Zbl 1043.30009
[14] L.V. Kovalev, J. Onninen, On invertibility of Sobolev mappings, preprint, 2008, arXiv:0812.2350; L.V. Kovalev, J. Onninen, On invertibility of Sobolev mappings, preprint, 2008, arXiv:0812.2350 · Zbl 1236.30021
[15] Manfredi, J. J.; Villamor, E., An extension of Reshetnyak’s theorem, Indiana Univ. Math. J., 47, 3, 1131-1145 (1998) · Zbl 0931.30014
[16] Martio, O.; Rickman, S.; Väisälä, J., Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I, 488 (1971) · Zbl 0223.30018
[17] Onninen, J., Mappings of finite distortion: Minors of the differential matrix, Calc. Var. Partial Differential Equations, 21, 4, 335-348 (2004) · Zbl 1072.30016
[18] Rajala, K., The local homeomorphism property of spatial quasiregular mappings with distortion close to one, Geom. Funct. Anal., 15, 5, 1100-1127 (2005) · 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; K. Rajala, Reshetnyak’s theorem and the inner distortion, Pure Appl. Math. Q., in press, University of Jyväskylä preprint, No. 336, 2007 · Zbl 1244.30041
[20] K. Rajala, Remarks on the Iwaniec-Šverák conjecture, University of Jyväskylä preprint, No. 377, 2009; K. Rajala, Remarks on the Iwaniec-Šverák conjecture, University of Jyväskylä preprint, No. 377, 2009
[21] Rickman, S., Quasiregular Mappings (1993), Springer-Verlag: Springer-Verlag Berlin · Zbl 0796.30018
[22] Tang, Q., Almost-everywhere injectivity in nonlinear elasticity, Proc. Roy. Soc. Edinburgh Sect. A, 109, 1-2, 79-95 (1988) · Zbl 0656.73010
[23] Väisälä, J., Lectures on \(n\)-Dimensional Quasiconformal Mappings, Lecture Notes in Math., vol. 229 (1971), Springer-Verlag: Springer-Verlag Berlin · Zbl 0221.30031
[24] Zorich, V. A., M.A. Lavrentyev’s theorem on quasiconformal space maps, Mat. Sb. (N.S.), 74, 116, 417-433 (1967) · Zbl 0181.08701
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