Some aspects of the Floquet theory for the heat equation in a periodic domain

We treat the linear heat equation in a periodic waveguide Pi subset of R-d, with a regular enough boundary, by using the Floquet transform methods. Applying the Floquet transform F to the equation yields a heat equation with mixed boundary conditions on the periodic cell (omega) over bar of Pi, and we analyse the connection between the solutions of the two problems. The considerations involve a description of the spectral projections onto subspaces H-S is an element of L-2( Pi) corresponding certain spectral components. We also show that the translated Wannier functions form an orthonormal basis in H-S.