the boundary value problem for quasi-regular mappings in space. The main result of the paper shows that for any ε> 0 and any piecewise affine map ϕ ∈ W 1,n (Ω; R n) with |Dϕ(x) | n � Ldet Dϕ(x) for almost all x ∈ Ω there exists a map u ∈ W 1,n (Ω; R n) such that |Du(x)| n = L det Du(x) a.e. in Ω, u|∂Ω = ϕ, ‖u − ϕ ‖ L n (Ω) < ε. The author exploits the idea of Baire’s category method as presented in series of papers by Dacorogna and Marcellini. However, the theorems of Dacorogna and Marcellini cannot be directly applied to the results of the author since the sets involved are unbounded. Reviewed by M. Yu. Vasil ′ chi
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