Abstract Given any f, a locally finitely piecewise affine homeomorphism of Rᵈ Ω ⊂ R d onto Rᵈ Δ ⊂ R d (for d=3, 4 d = 3, 4) such that f W^1, p (, Rᵈ) f ∈ W 1, p (Ω, R d) and f^-1 W^1, q (, Rᵈ) f - 1 ∈ W 1, q (Δ, R d), 1 p, q 1 ≤ p, q ∞ and any >0 ε > 0 we construct a diffeomorphism f f ~ such that aligned f-f ₖ^₁, (, {Rᵈ) } + f^-1-f^-1 ₖ^₁, ₐ (, {Rᵈ) } ‖ f - f ~ ‖ W 1, p (Ω,
Campbell et al. (Tue,) studied this question.