A density result for Sobolev spaces in dimension two, and applications to stability of nonlinear Neumann problems

We prove that if $\text{Om}\subseteq {\mathbb{R}}^2$ is bounded and ${\mathbb{R}}^2\setminus \text{Om}$ satisfies suitable structural assumptions (for example it has a countable number of connected components), then $W^{1,2}(\text{Om})$ is dense in $W^{1,p}(\text{Om})$ 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 $$\{\begin{array}{ll}-\mathrm{d}\mathrm{i}\mathrm{v}A(x,\mathrm{\nabla }u)+B(x,u)=0& \text{in}\text{Om},\\ A(x,\mathrm{\nabla }u)\cdot \nu =0& \text{on}\mathrm{\partial }\text{Om},\end{array}$$ where $A:{\mathbb{R}}^2\times {\mathbb{R}}^2\to {\mathbb{R}}^2$ and $B:{\mathbb{R}}^2\times \mathbb{R}\to \mathbb{R}$ are Carath'eodory functions which satisfy standard monotonicity and growth conditions of order $p$.