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 \(\Omega'\).