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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.)

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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.

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  • $\begingroup$ Thanks. I know these references and (I may be wrong) I didn’t think that they were relevant. They consider developable surfaces, whereas in my problem the mapping is any W22 function. On the other hand, they are looking for smooth isometric approximations, whereas I don’t even have a metric on the manifold, so in this respect I have less constraints. $\endgroup$
    – Raz Kupferman
    Commented Oct 28, 2019 at 18:27
  • $\begingroup$ My question made no sense as formulated. I meant that $df$ has full rank a.e. $\endgroup$
    – Raz Kupferman
    Commented Oct 28, 2019 at 19:45
  • $\begingroup$ @RazKupferman Sure. My comment made so sense as well so I deleted it. I like the problem and I will think about it. $\endgroup$
    – Piotr Hajlasz
    Commented Oct 28, 2019 at 19:54

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