First order operators on shrinking graph-like spaces

TL;DR

This paper proves that first order differential operators on graph-like spaces converge to those on the underlying metric graph, under certain geometric and uniformity conditions, with implications for Laplace operators.

Contribution

It establishes generalized norm resolvent convergence of first order operators from graph-like spaces to metric graphs, extending previous understanding of operator limits in geometric analysis.

Findings

Proves convergence of first order operators on graph-like spaces to metric graph operators.

Establishes a uniform Gaffney estimate using divergence and curvature relations.

Demonstrates convergence results under convexity and connectivity assumptions.

Abstract

In this article we discuss the convergence of first order operators on a thickened graph (a graph-like space) towards a similar operator on the underlying metric graph. On the graph-like space, the first order operator is of the form exterior derivative (the gradient) on functions and its adjoint (the negative divergence) on closed 1-forms (irrotational vector fields). Under the assumption that each cross section of the tubular edge neighbourhood is convex, that each vertex neighbourhood is simply connected and under suitable uniformity assumptions (which hold in particular, if the spaces are compact) we establish generalised norm resolvent convergence of the first order operator on the graph-like space towards the one on the metric graph. The square of the first order operator is of Laplace type; on the metric graph, the function (0-form) component is the usual standard (Kirchhoff) Laplacian. A key ingredient in the proof is a uniform Gaffney estimate: such an estimate follows from an equality relating here the divergence operator with all (weak) partial derivatives and a curvature term, together with a (localised) Sobolev trace estimate.