  We study second-order divergence-form systems on half-infinite cylindrical
domains with a bounded and possibly rough base, subject to homogeneous mixed
boundary conditions on the lateral boundary and square integrable Dirichlet,
Neumann, or regularity data on the cylinder base. Assuming that the
coefficients A are close to coefficients A\_0 that are independent of the
unbounded direction with respect to the modified Carleson norm of Dahlberg, we
prove a priori estimates and establish well-posedness if A\_0 has a special
structure. We obtain a complete characterization of weak solutions whose
gradient either has an L^2-bounded non-tangential maximal function or satisfies
a Lusin area bound. Our method relies on the first-order formalism of Axelsson,
McIntosh, and the first author and the recent solution of Kato's conjecture for
mixed boundary conditions due to Haller-Dintelmann, Tolksdorf, and the second
author.
