A note on non-homogeneous hyperbolic operators with low-regularity coefficients

TL;DR

This paper derives an energy estimate for a class of hyperbolic operators with coefficients that have low regularity, specifically satisfying log-Zygmund and log-Lipschitz conditions in time and space.

Contribution

It provides a novel energy estimate for hyperbolic operators with low-regularity coefficients under specific continuity conditions.

Findings

Established energy estimates for hyperbolic operators with low-regularity coefficients.

Extended analysis to coefficients with log-Zygmund and log-Lipschitz continuity.

Results applicable to operators with minimal regularity assumptions.

Abstract

In this paper we obtain an energy estimate for a complete strictly hyperbolic operator with second order coefficients satisfying a log-Zygmund-continuity condition with respect to $t$, uniformly with respect to $x$, and a log-Lipschitz-continuity condition with respect to $x$, uniformly with respect to $t$.