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

On bounded domains Ω⊂ R³, 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 W1,q_Γ(Ω) to W-1,q_Γ(Ω) for some $q>3$. We indicate relevant examples from real-world applications.