On the density of classes of closed convex sets with pointwise constraints in Sobolev spaces

For a Banach space X   of RMRM-valued functions on a Lipschitz domain, let K(X)⊂XK(X)⊂X be a closed convex set arising from pointwise constraints on the value of the function, its gradient or its divergence, respectively. The main result of the paper establishes, under certain conditions, the density of K(X0)K(X0) in K(X1)K(X1) where X0X0 is densely and continuously embedded in X1X1. The proof is constructive, utilizes the theory of mollifiers and can be applied to Sobolev spaces such as H0(div,Ω)H0(div,Ω) and W01,p(Ω), in particular. It is also shown that such a density result cannot be expected in general.