Traces for Hilbert Complexes

TL;DR

This paper develops a general theory of trace operators and spaces for Hilbert complexes, extending classical boundary trace concepts to an abstract setting with applications to the de Rham complex.

Contribution

It introduces a new framework for trace spaces in Hilbert complexes, characterizes their properties, and links them to classical boundary traces in Euclidean domains.

Findings

Trace spaces are characterized as quotient spaces and annihilators.

Many properties of classical boundary traces are derived from the abstract Hilbert complex structure.

Under stable regular decompositions, the trace Hilbert complex is shown to be Fredholm.

Abstract

We study a new notion of trace operators and trace spaces for abstract Hilbert complexes. We introduce trace spaces as quotient spaces/annihilators. We characterize the kernels and images of the related trace operators and discuss duality relationships between trace spaces. We elaborate that many properties of the classical boundary traces associated with the Euclidean de Rham complex on bounded Lipschitz domains are rooted in the general structure of Hilbert complexes. We arrive at abstract trace Hilbert complexes that can be formulated using quotient spaces/annihilators. We show that, if a Hilbert complex admits stable "regular decompositions" with compact lifting operators, then the associated trace Hilbert complex is Fredholm. Incarnations of abstract concepts and results in the concrete case of the de Rham complex in three-dimensional Euclidean space will be discussed throughout.