Elliptic complexes of first-order cone operators: ideal boundary conditions

TL;DR

This paper characterizes the $L^2$ domains of first-order elliptic complexes on manifolds with conical singularities, analyzing boundary conditions and domain structures to understand their Hilbert complex properties.

Contribution

It provides a detailed description of $L^2$ domain spaces for elliptic complex operators on conical manifolds, including conditions for domain mappings and the structure of domain choices.

Findings

Nondegeneracy of a pairing of cohomology classes is established.

The set of domain choices forms an algebraic variety.

A necessary and sufficient condition for domain mapping is proven.

Abstract

The purpose of this paper is to provide a detailed description of the spaces that can be specified as $L^2$ domains for the operators of a first order elliptic complex on a compact manifold with conical singularities. This entails an analysis of the nature of the minimal domain and of a complementary space in the maximal domain of each of the operators. The key technical result is the nondegeneracy of a certain pairing of cohomology classes associated with the indicial complex. It is further proved that the set of choices of domains leading to Hilbert complexes in the sense of Br\"uning and Lesch form a variety, as well as a theorem establishing a necessary and sufficient condition for the operator in a given degree to map its maximal domain into the minimal domain of the next operator.