A continuum limit for the Laplace operator on metric graphs

Sidney Holden, Geoffrey Vasil

Published 2023-01-17Version 1

Metric graphs -- with continuous real-valued edges -- have been studied extensively as domains for solutions of PDEs. We study the action of the Laplace operator on each edge as a metric graph fills in a $n-$dimensional Riemannian manifold, e.g., a very dense spider web in the planar disc or ball of wire within the $2-$sphere. We derive an $n$-dimensional limiting differential operator whose low-order eigenmodes approximate the low-order eigenmodes of the metric graph Laplace operator. High-density metric graphs exhibit nontrivial, continuum-like behaviour at low frequencies and purely graph-like behaviour at high frequencies. In the continuum limit, we find the emergence of a new symmetric tensor field but with an unusual distance scaling compared to the traditional Riemannian metric. The limiting operator from a metric graph implies a possible new kind of continuous geometry.