Optimal Sobolev Regularity for Linear Second-Order Divergence Elliptic Operators Occurring in Real-World Problems

On bounded domains Ω ⊂ R, we consider divergence-type operators −∇ · μ∇, including mixed homogeneous Dirichlet and Neumann boundary conditions on ∂Ω \ Γ and Γ ⊂ ∂Ω, respectively, and discontinuous coefficient functions μ. We develop a general geometric framework for Ω, Γ and μ in which it is possible to prove that −∇ · μ∇ + 1 provides an isomorphism from W 1,q Γ (Ω) to W Γ (Ω) for some q > 3. We indicate relevant examples from real-world applications.