  We consider operator-valued boundary value problems in $(0,2\pi)^n$ with
periodic or, more generally, $\nu$-periodic boundary conditions. Using the
concept of discrete vector-valued Fourier multipliers, we give equivalent
conditions for the unique solvability of the boundary value problem. As an
application, we study vector-valued parabolic initial boundary value problems
in cylindrical domains $(0,2\pi)^n\times V$ with $\nu$-periodic boundary
conditions in the cylindrical directions. We show that under suitable
assumptions on the coefficients, we obtain maximal $L^q$-regularity for such
problems.
