  In this paper we present a general extrapolated elliptic regularity result
for second order differential operators in divergence form on fractional
Sobolev-type spaces of negative order $X^{s-1,q}_D(\Omega)$ for $s > 0$ small,
including mixed boundary conditions and with a fully nonsmooth geometry of
$\Omega$ and the Dirichlet boundary part $D$. We expect the result to find
applications in the analysis of nonlinear parabolic equations, in particular
for quasilinear problems or when treating coupled systems of equations. To
demonstrate the usefulness of our result, we give a new proof of local-in-time
existence and uniqueness for the van Roosbroeck system for semiconductor
devices which is much simpler than already established proofs.
