Regularity properties of isometric immersions

We show that an isometric immersion y from a two-dimensional domain S with C1,α boundary to ℝ3 which belongs to the critical Sobolev space W2,2 is C1 up to the boundary. More generally C1 regularity up to the boundary holds for all scalar functions V ∈ W2,2(S) which satisfy det ∇2V=0. If S has only Lipschitz boundary we show such V can be approximated in W2,2 by functions Vk ∈ W1,∞∩W2,2 with det ∇2Vk=0.