On Lars H\"ormander's remark on the characteristic Cauchy problem

TL;DR

This paper extends Hörmander's 1990 results on solving the characteristic Cauchy problem for second order wave equations, reducing regularity assumptions on the metric and potentials while maintaining solution existence.

Contribution

It generalizes previous work by lowering regularity requirements on spacetime metrics and potentials for solving the characteristic Cauchy problem.

Findings

Solved the spacelike initial data Cauchy problem with Lipschitz metric and bounded potentials.

Addressed the fully characteristic Cauchy problem with slightly more regular metrics and potentials.

Maintained finite energy solution spaces similar to the smooth case.

Abstract

We extend the results of a work by L. H\"ormander in 1990 concerning the resolution of the characteristic Cauchy problem for second order wave equations with regular first order potentials. The geometrical background of this work was a spatially compact spacetime with smooth metric. The initial data surface was spacelike or null at each point and merely Lipschitz. We lower the regularity hypotheses on the metric and potential and obtain similar results. The Cauchy problem for a spacelike initial data surface is solved for a Lipschitz metric and coefficients of the first order potential that are $L^\infty_\mathrm{loc}$, with the same finite energy solution space as in the smooth case. We also solve the fully characteristic Cauchy problem with very slightly more regular metric and potential, namely a ${\cal C}^1$ metric and a potential with continuous first order terms and locally $L^\infty$ coefficients for the terms of order 0.