From Whole-Space Theory to Boundary Value Problems for Mixed-Order PDEs

Abstract

We develop a general principle for passing from the analysis of partial differential operators on the whole space to boundary value problems on the half-space. The reduction separates the boundary problem into restricted invertibility of the interior operator and a complementing problem formulated entirely on the trace space. For a broad class of operators admitting suitable factorizations, the restricted invertibility is governed by the whole-space theory of the factors, while the complementing problem determines both well-posedness and the admissible combinations of interior and boundary data. In this way, boundary value theory is reduced to whole-space operator theory together with an explicit analysis on the trace space. We realize this principle for time-periodic partial differential equations and systems of mixed order, developing the required Newton-polygon function spaces, interpolation theory, and $L^p$-trace theory. For mixed-order differential operators under general boundary conditions, well-posedness and the admissible data space are determined by an explicit complemented boundary matrix. This yields new $L^p$-well-posedness results for Cahn--Hilliard equations with dynamic boundary conditions and for Cahn--Hilliard--Gurtin systems. Most notably, we obtain the first maximal $L^p$-regularity result for a genuinely mixed-order interior system coupled to dynamic boundary conditions carrying a distinct mixed-order structure.

arXiv commentSubstantial reorganization emphasizing the abstract reduction from whole-space theory to boundary value problems; streamlined and shortened the interpolation and trace arguments; added applications to Cahn--Hilliard equations with Liu--Wu boundary conditions and to the Cahn--Hilliard--Gurtin system with dynamic Goldstein--Miranville boundary conditions; corrected typos