On bounded domains Ω ⊂ R 3 , 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 −1,q Γ (Ω) for some q > 3. We indicate relevant examples from real-world applications.

Interpolation theorems are proved for Sobolev spaces of functions on nonsmooth domains with vanishing trace on a part of the boundary.

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We consider a stationary Schrödinger-Poisson system on a bounded interval of the real axis. The Schrödinger operator is defined on the bounded domain with transparent boundary conditions. This allows to model a non-zero current flow trough the boundary of the interval. We prove that the system always admits a solution and give explicit a priori estimates for the solutions.

Let Υ be a three-dimensional Lipschitz polyhedron, and assume that the matrix function µ is piecewise constant on a polyhedral partition of Υ. Based on regularity results for solutions to two-dimensional anisotropic transmission problems near corner points we obtain conditions on µ and the intersection angles between interfaces and ∂Υ ensuring that the operator −∇ · µ∇ maps the Sobolev space W 1,q 0 (Υ) isomorphically onto W −1,q (Υ) for some q > 3.