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

We prove that if Omega subset of R-2 is bounded and R-2 \ Omega satisfies suitable structural assumptions (for example it has a countable number of connected components), then W-1,W-2 (Omega) is dense in W-1,W-p (Omega) 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 {-div A(x, del u) + B(x, u) = 0 in Omega, A(x, del u) center dot v = 0 on partial derivative Omega, where A : R-2 x R-2 -> R-2 and B : R-2 x R -> R are Caratheodory functions which satisfy standard monotorricity and growth conditions of order p. (C) 2007 Elsevier Inc. All rights reserved.