A Density Result for Sobolev Spaces in Dimension Two, and Applications to Stability of Nonlinear Neumann Problems
نویسنده
چکیده
We prove that if Ω ⊆ R2 is bounded and R2 \Ω satisfies suitable structural assumptions (for example it has a countable number of connected components), then W 1,2(Ω) is dense in W 1,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 ( −divA(x,∇u) +B(x, u) = 0 in Ω, A(x,∇u) · ν = 0 on ∂Ω, where A : R2×R2 → R2 and B : R2×R → R are Carathéodory functions which satisfy standard monotonicity and growth conditions of order p.
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