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 Lp-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 Lp-well-posedness results for Cahn--Hilliard equations with dynamic boundary conditions and for Cahn--Hilliard--Gurtin systems. Most notably, we obtain the first maximal Lp-regularity result for a genuinely mixed-order interior system coupled to dynamic boundary conditions carrying a distinct mixed-order structure.
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BibTeX
@misc{neuttiens2026from,
title = {{From Whole-Space Theory to Boundary Value Problems for Mixed-Order PDEs}},
author = {Guillaume Neuttiens and Jonas Sauer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.12250},
eprint = {2608.12250},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.12250},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From Whole-Space Theory to Boundary Value Problems for Mixed-Order PDEs
AU - Guillaume Neuttiens
AU - Jonas Sauer
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2608.12250
UR - https://arxiv.org/abs/2608.12250
AB - 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.
ER -