TITLE: Approximation of a Sobolev surface by a smooth surface
QUESTION [6 upvotes]: 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 $Df$ has full rank 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.)

REPLY [2 votes]: I do not know the full answer yet, but in the case in which $f:\Omega\to\mathbb{R}^3$, $\Omega\subset\mathbb{R}^2$ bounded and convex, is an isometric immersion, then it can be approximated by smooth isometric immersions in $W^{2,2}$ norm. This is Theorem I in:
M. R. Pakzad, 
On the Sobolev space of isometric immersions.
J. Differential Geom. 66 (2004), no. 1, 47–69. 
Perhaps you can find relevant references and results there. The result of Pakzad has been generalized in:
Z. Liu, M. R. Pakzad, Rigidity and regularity of codimension-one Sobolev isometric immersions. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 14 (2015), no. 3, 767–817.