On mixed initial–boundary value problems for systems that are not strictly hyperbolic

Abstract

The classical theory of strictly hyperbolic boundary value problems has received several extensions since the 70s. One of the most noticeable is the result of Metivier establishing Majda’s “block structure condition” for constantly hyperbolic operators, which implies well-posedness for the initial–boundary value problem (IBVP) with zero initial data. The well-posedness of the IBVP with non-zero initial data requires that “L2 is a continuable initial condition”. For strictly hyperbolic systems, this result was proven by Rauch. We prove here, by using classical matrix theory, that his fundamental a priori estimates are valid for constantly hyperbolic IBVPs.