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

We prove that if Ω⊆R2Ω⊆R2 is bounded and R2∖ΩR2∖Ω satisfies suitable structural assumptions (for example it has a countable number of connected components), then W1,2(Ω)W1,2(Ω) is dense in W1,p(Ω)W1,p(Ω) for every 1⩽p<21⩽p<2. The main application of this density result is the study of stability under boundary variations for nonlinear Neumann problems of the form{−divA(x,∇u)+B(x,u)=0in Ω,A(x,∇u)⋅ν=0on ∂Ω, where A:R2×R2→R2 and B:R2×R→R are Carathéodory functions which satisfy standard monotonicity and growth conditions of order p.