We consider the initial-boundary value problem for systems of quasilinear wave equations on domains of the form $[0,T] times Sigma$, where $Sigma$ is a compact manifold with smooth boundaries $partialSigma$. By using an appropriate reduction to a fir
st order symmetric hyperbolic system with maximal dissipative boundary conditions, well posedness of such problems is established for a large class of boundary conditions on $partialSigma$. We show that our class of boundary conditions is sufficiently general to allow for a well posed formulation for different wave problems in the presence of constraints and artificial, nonreflecting boundaries, including Maxwells equations in the Lorentz gauge and Einsteins gravitational equations in harmonic coordinates. Our results should also be useful for obtaining stable finite-difference discretizations for such problems.
In recent work, we used pseudo-differential theory to establish conditions that the initial-boundary value problem for second order systems of wave equations be strongly well-posed in a generalized sense. The applications included the harmonic versio
n of the Einstein equations. Here we show that these results can also be obtained via standard energy estimates, thus establishing strong well-posedness of the harmonic Einstein problem in the classical sense.
