Finite Propagation Speed for First Order Systems and Huygens' Principle for Hyperbolic Equations

TL;DR

This paper establishes finite propagation speed for first order systems on Riemannian manifolds and extends the weak Huygens' principle to second order hyperbolic equations, offering new proofs and broader applicability.

Contribution

It provides a new direct proof for finite propagation speed in self-adjoint systems and extends the results to operators on metric measure spaces.

Findings

Finite propagation speed holds for strongly continuous groups generated by first order systems.

A new approach to the weak Huygens' principle for second order hyperbolic equations is developed.

Extension of finite propagation speed results to metric measure spaces.

Abstract

We prove that strongly continuous groups generated by first order systems on Riemannian manifolds have finite propagation speed. Our procedure provides a new direct proof for self-adjoint systems, and allows an extension to operators on metric measure spaces. As an application, we present a new approach to the weak Huygens' principle for second order hyperbolic equations.