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

We treat the linear heat equation in a periodic waveguide Π ⊂ Rᵈ Π ⊂ R d , with a regular enough boundary, by using the Floquet transform methods. Applying the Floquet transform F F to the equation yields a heat equation with mixed boundary conditions on the periodic cell ϖ of Π Π , and we analyse the connection between the solutions of the two problems. The considerations involve a description of the spectral projections onto subspaces HS ⊂ L²(Π ) H S ⊂ L 2 ( Π ) corresponding certain spectral components. We also show that the translated Wannier functions form an orthonormal basis in HS H S .