W1,q regularity results for elliptic transmission problems on heterogeneous polyhedra

Let $\Upsilon$ be a three-dimensional Lipschitz polyhedron, and assume that the matrix function $\mu$ is piecewise constant on a polyhedral partition of $\Upsilon$. Based on regularity results for solutions to two-dimensional anisotropic transmission problems near corner points we obtain conditions on $\mu$ and the intersection angles between interfaces and $\partial \Upsilon$ ensuring that the operator $-\nabla \cdot \mu \nabla$ maps the Sobolev space $W^{1,q}_0(\Upsilon)$ isomorphically onto $W^{-1,q}(\Upsilon)$ for some $q > 3$