Discrete Fourier multipliers and cylindrical boundary-value problems

Abstract

We consider operator-valued boundary-value problems in (0, 2π)n with periodic or, more generally, ν-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, 2π)n × V with ν-periodic boundary conditions in the cylindrical directions. We show that, under suitable assumptions on the coefficients, we obtain maximal Lq-regularity for such problems. For symmetric operators such as the Laplacian, related results for mixed Dirichlet-Neumann boundary conditions on (0, 2π)n × V are deduced.