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Let Ω ⊆ R 2 be a domain and let f ∈ W 1,1 (Ω, R 2 ) be a homeomorphism (between Ω and f (Ω)). Then there exists a sequence of smooth diffeomorphisms f k converging to f in W 1,1 (Ω, R 2 ) and uniformly.2000 Mathematics Subject Classification. 46E35.

Let Ω ⊂ R n , n ≥ 4, be a domain and 1 ≤ p < [n/2], where [a] stands for the integer part of a. We construct a homeomorphism f ∈ W 1,p ((−1, 1) n , R n ) such that J f = det Df > 0 on a set of positive measure and J f < 0 on a set of positive measure. It follows that there are no diffeomorphisms (or piecewise affine homeomorphisms) f k such that f k → f in W 1,p .

Let Ω ⊂ Rn be open. Given a homeomorphism of finite distortion with |Df| in the Lorentz space Ln−1, 1 (Ω), we show that and f−1 has finite distortion. A class of counterexamples demonstrating sharpness of the results is constructed.

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Suppose that f = (u, v) is a homeomorphism in the plane of the Sobolev class W 1,1 loc such that its inverse is of the same Sobolev class. We prove that u and v have the same set of critical points. As an application we show that u and v are distributional solutions to the same non-trivial degenerate elliptic equation in divergence form. We study similar properties also in higher dimensions.