An old important question is to approximate W1, 2 orientation-preserving homeomorphisms by means of smooth ones, or at least of piecewise affine ones. This is particularly important in the context of non-linear elasticity.

However, this is not easy to do, since classical results only allow to approximate with piecewise affine functions, but in the L8 sense instead of the W1, 2 sense that one would need.

In this talk, we will describe some history of the problem and the state-of-the-art, then we will present some very recent contributions.