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

Abstract

We treat the linear heat equation in a periodic waveguide \(\Pi \subset {{\mathbb {R}}}^d\), with a regular enough boundary, by using the Floquet transform methods. Applying the Floquet transform \({{\textsf{F}}}\) to the equation yields a heat equation with mixed boundary conditions on the periodic cell \(\varpi \) 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 \({{\mathcal {H}}}_S \subset L^2(\Pi )\) corresponding certain spectral components. We also show that the translated Wannier functions form an orthonormal basis in \({{\mathcal {H}}}_S\).