A Carleman estimate and an energy method for a first-order symmetric hyperbolic system

TL;DR

This work derives a Carleman estimate for a first-order symmetric hyperbolic system with symmetric coefficient matrices under a positivity condition on the combined weight, and leverages it to obtain an observability inequality that bounds the initial state by boundary measurements. The approach introduces a weighted transform $w=e^{s\varphi}u$, compensates cross terms via a matrix multiplier, and utilizes a rescaled parameterization to ensure coercivity, allowing absorption of lower-order terms. By coupling the Carleman estimate with an energy method, the authors show that initial data can be recovered from boundary data on $\partial\Omega\times(0,T)$, under the assumptions (1.6)-(1.8) and a suitable choice of $\eta$ and $\beta$. The results extend Carleman-type techniques to non-diagonal first-order systems and contribute to unique continuation and inverse problems for hyperbolic PDEs, with potential applications to Maxwell and elasticity models.

Abstract

For a symmetric hyperbolic system of the first order, we prove a Carleman estimate under some positivity condition concerning the coefficient matrices. Next, applying the Carleman estimate, we prove an observability $L^2$-estimate for initial values by boundary data.