We prove that every W2,2 isometric immersion from a convex regular domain of ℝ2 into ℝ3 can be approximated in W2,2-norm by smooth isometric immersions from the same domain into ℝ3.
"On the Sobolev Space of Isometric Immersions." J. Differential Geom. 66 (1) 47 - 69, January, 2004. https://doi.org/10.4310/jdg/1090415029