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

We prove that if $A$ is bounded open subset in the plane and its complement satisfies suitable structural assumptions (for example it has a countable number of connected components), then $W^{1,2}(A)$ is dense in $W^{1,p}(A)$ for every exponent p between 1 and 2. The main application of this density result is the study of stability under boundary variations for nonlinear elliptic problems under Neumann conditions.