  We prove that if $\Om \subseteq \R^2$ is bounded and $\R^2 \setminus \Om$
satisfies suitable structural assumptions (for example it has a countable
number of connected components), then $W^{1,2}(\Om)$ is dense in $W^{1,p}(\Om)$
for every $1\le p<2$. The main application of this density result is the study
of stability under boundary variations for nonlinear Neumann problems of the
form $$ \begin{cases} -{\rm div} A(x,\nabla u)+B(x,u)=0 & \text{in}\Om, \\
A(x,\nabla u)\cdot \nu=0 & \text{on}\partial \Om, \end{cases} $$ where
$A:\R^2\times \R^2 \to \R^2$ and $B:\R^2 \times \R \to \R$ are Carath\'eodory
functions which satisfy standard monotonicity and growth conditions of order
$p$.
