Vodop’yanov, S. K. Regularity of mappings inverse to Sobolev mappings. (English. Russian original) Zbl 1266.26019 Sb. Math. 203, No. 10, 1383-1410 (2012); translation from Mat. Sb. 203, No. 10, 3-32 (2012). The paper addresses the issue of the regularity of Sobolev homeomorphisms \(\varphi:\Omega\to\Omega'\) in an Euclidean space, or, more generally, of weakly differentiable homeomorphisms. This is a topic of substantial current interest, with recent contributions such as S. Hencl, P. Koskela and J. Malý, [Proc. R. Soc. Edinb., Sect. A, Math. 136, No. 6, 1267–1285 (2006; Zbl 1122.30015)], S. Hencl, P. Koskela and J. Onninen, [Arch. Ration. Mech. Anal. 186, No. 3, 351–360 (2007; Zbl 1155.26007)] and M. Csörnyei, S. Hencl and J. Malý, [J. Reine Angew. Math. 644, 221–235 (2010; Zbl 1210.46023)]. In general, the weaker the regularity imposed on \(\varphi\), the more subtle such inverse mapping theorems become. One of the crucial assumptions in the present paper is finite codistortion, which stipulates that the adjugate \(\mathrm{adj}\, D\varphi\) of the differential matrix \(D\varphi\) is zero a.e. on the set where the Jacobian determinant \(\det D\varphi\) is zero.A representative result is Theorem 2. Suppose that (1) \(\varphi\) is approximately differentiable a.e.; (2) \(D\varphi\) is in \(L^1\); (3) \(\varphi\) has finite codistortion; (4) \(\varphi\) has Luzin’s property (N) with respect to the \((n-1)\)-dimensional measure on almost every coordinate-parallel hyperplane crossing \(\Omega\). Then the inverse map \(\varphi^{-1}\) satisfies the following: (5) \(\varphi^{-1}\) is of class ACL; (6) its derivative \(D\varphi^{-1}\) is in \(L^1\); (7) \(\varphi^{-1}\) has finite distortion. Conversely, if \(\varphi^{-1}\) satisfies (5)–(7) and \(\varphi\) is approximately differentiable on the zero set of the volume Jacobian of \(\varphi\), then \(\varphi\) has properties (1)–(4).The author also investigates the composition operators that homeomorphisms induce between Sobolev spaces on \(\Omega\) and \(\Omega'\). Reviewer: Leonid Kovalev (Syracuse) Cited in 1 ReviewCited in 33 Documents MSC: 26B10 Implicit function theorems, Jacobians, transformations with several variables 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations Keywords:Sobolev homeomorphism; inverse mapping; approximate differentiability; composition operator Citations:Zbl 1122.30015; Zbl 1155.26007; Zbl 1210.46023 PDFBibTeX XMLCite \textit{S. K. Vodop'yanov}, Sb. Math. 203, No. 10, 1383--1410 (2012; Zbl 1266.26019); translation from Mat. Sb. 203, No. 10, 3--32 (2012) Full Text: DOI