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 \({{\mathcal {H}}}_S\).

我们通过使用 Floquet 变换方法，以足够规则的边界来处理周期性波导\(\Pi \subset {{\mathbb {R}}}^d\)中的线性热方程。对方程应用 Floquet 变换\({{\textsf{F}}}\)可以得到在\(\Pi \)的周期单元\(\varpi \)上具有混合边界条件的热方程，我们分析了两个问题的解决方案之间的联系。这些考虑因素涉及到对应某些光谱分量的子空间\({{\mathcal {H}}}_S \subset L^2(\Pi )\)上的光谱投影的描述。我们还表明，转换后的 Wannier 函数在\({{\mathcal {H}}}_S\)中形成了正交基。