Green operators for low regularity spacetimes

TL;DR

This paper develops Green operators for wave equations on low-regularity spacetimes, ensuring well-posedness and providing explicit kernel formulas, advancing the mathematical understanding of wave propagation in less smooth geometries.

Contribution

It introduces a method to construct Green operators on low-regularity spacetimes and derives explicit kernel formulas using eigenbasis and Green matrices.

Findings

Green operators are well-defined under generalized hyperbolicity.

Explicit kernel formulas are derived for these Green operators.

The approach extends wave analysis to less regular spacetime geometries.

Abstract

In this paper we define and construct advanced and retarded Green operators for the wave operator on spacetimes with low regularity. In order to do so we require that the spacetime satisfies the condition of generalised hyperbolicity which is equivalent to well- posedness of the classical inhomogeneous problem with zero initial data where weak solutions are properly supported. Moreover, we provide an explicit formula for the kernel of the Green operators in terms of an arbitrary eigenbasis of H 1 and a suitable Green matrix that solves a system of second order ODEs.