The Time-Periodic Cahn–Hilliard–Gurtin System on the Half Space as a Mixed-Order System with General Boundary Conditions

A well-posedness and maximal regularity result for the time-periodic Cahn–Hilliard–Gurtin system in the half space is proved. For this purpose, we introduce a novel class of complementing boundary conditions, extending the classical Lopatinskiĭ–Shapiro conditions from elliptic and parabolic theory to time-periodic mixed-order systems with general boundary conditions. Moreover, we show that the classical Lopatinskiĭ–Shapiro conditions are in general insufficient for well-posedness of mixed-order systems.