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

TL;DR

This paper establishes well-posedness and maximal regularity for the time-periodic Cahn-Hilliard-Gurtin system in the half space, introducing new boundary conditions extending classical theories to mixed-order systems.

Contribution

It introduces a novel class of complementing boundary conditions for mixed-order systems, extending classical Lopatinski-Shapiro conditions to time-periodic problems.

Findings

Proves well-posedness and maximal regularity for the system.

Develops a new class of boundary conditions for mixed-order systems.

Shows classical conditions are insufficient for such systems.

Abstract

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 Lopatinski\u{\i}-Shapiro conditions from elliptic and parabolic theory to time-periodic mixed-order systems with general boundary conditions. Moreover, we show that the classical Lopatinski\u{\i}-Shapiro conditions are in general insufficient for well-posedness of mixed-order systems.