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

We prove that if is bounded and satisfies suitable structural assumptions (for example it has a countable number of connected components), then W1,2(Ω) is dense in W1,p(Ω) for every 1 p<2. The main application of this density result is the study of stability under boundary variations for nonlinear Neumann problems of the form where and are Carathéodory functions which satisfy standard monotonicity and growth conditions of order p.