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 $$\Pi \subset {{\mathbb {R}}}^d$$ Π ⊂ R d , with a regular enough boundary, by using the Floquet transform methods. Applying the Floquet transform $${{\textsf{F}}}$$ F to the equation yields a heat equation with mixed boundary conditions on the periodic cell $$\varpi $$ ϖ of $$\Pi $$ Π , and we analyse the connection between the solutions of the two problems. The considerations involve a description of the spectral projections onto subspaces $${{\mathcal {H}}}_S \subset L^2(\Pi )$$ H S ⊂ L 2 ( Π ) corresponding certain spectral components. We also show that the translated Wannier functions form an orthonormal basis in $${{\mathcal {H}}}_S$$ H S .