Finite propagation speed for solutions of the wave equation on metric graphs

TL;DR

This paper establishes a class of self-adjoint Laplace operators on metric graphs ensuring solutions to the wave equation exhibit finite propagation speed, using energy methods adapted from smooth manifold analysis.

Contribution

It introduces a new class of Laplace operators on metric graphs that guarantee finite propagation speed for wave solutions, extending energy method techniques.

Findings

Solutions have finite propagation speed under the new operators.

Energy methods from smooth manifolds are successfully adapted.

Provides a framework for analyzing wave equations on metric graphs.

Abstract

We provide a class of self-adjoint Laplace operators on metric graphs with the property that the solutions of the associated wave equation satisfy the finite propagation speed property. The proof uses energy methods, which are adaptions of corresponding methods for smooth manifolds.