An $L^p$-Theory for Time-Periodic Mixed-Order Partial Differential Equations under General Boundary Conditions

Abstract

We develop an $L^p$-theory for time-periodic boundary value problems associated with partial differential equations and systems of mixed order. Our approach is based on anisotropic function spaces described by order functions and their associated Newton polygons. We develop a trace theory on the half-space, including a characterization of the trace spaces by real interpolation. For general boundary conditions, we introduce an abstract framework in which well-posed\-ness is characterized by a complementing condition formulated in terms of the traces of solutions. In particular, the admissible data space, including the compatibility conditions induced by the boundary operators, emerges naturally from the abstract framework. For mixed-order differential operators, the complementing condition is reduced to the invertibility of a complemented boundary matrix, yielding an explicit criterion for well-posedness in $L^p$-based spaces. The resulting theory applies to general Newton polygon structures and allows for boundary operators involving time derivatives. As applications, we establish time-periodic $L^p$-well-posedness for the Cahn--Hilliard--Gurtin system and for parabolic problems with dynamic boundary conditions.