I was quite sure that the answer to the following question is known, and was surprised not to find any reference:

Let $M$ be a compact, oriented $2$-dimensional manifold with boundary. Let $f:M\to R^3$ be a $W^{2,2}$-map such that $\det df>0$ a.e. Can $f$ be approximated in $W^{2,2}$ by smooth immersions? (As standard, one can endow $M$ with any Riemannian metric to define the Sobolev spaces.)