Layer potentials C*-algebras of domains with conical points

TL;DR

This paper constructs and analyzes a C*-algebra associated with domains having conical points, using groupoid theory and pseudodifferential operators to understand boundary value problems with complex geometries.

Contribution

It introduces an explicit groupoid model for the boundary of such domains and computes the algebra's K-groups, providing new tools for boundary value problems with conical singularities.

Findings

Constructed a groupoid associated to domain boundaries with conical points.

Computed the K-theory of the associated C*-algebra.

Established Fredholm conditions for pseudodifferential operators in this setting.

Abstract

To a domain with conical points \Omega, we associate a natural C*-algebra that is motivated by the study of boundary value problems on \Omega, especially using the method of layer potentials. In two dimensions, we allow \Omega to be a domain with ramified cracks. We construct an explicit groupoid associated to the boundary of \Omega and use the theory of pseudodifferential operators on groupoids and its representations to obtain our layer potentials C*-algebra. We study its structure, compute the associated K-groups, and prove Fredholm conditions for the natural pseudodifferential operators affiliated to this C*-algebra.