Elliptic-hyperbolic systems and the Einstein equations
The authors of this interesting paper study the Einstein equations in a gauge given by a combination of the constant mean curvature and spatial harmonic coordinate conditions. This leads to a coupled quasi-linear elliptic-hyperbolic system of evolution equations. The hyperbolic part of this system is a modified version of the Einstein evolution equations, and the elliptic part consists of the defining equations for lapse and shift. The authors prove that the Cauchy problem is locally strongly well posed and that a continuation principle holds. For initial data satisfying the Einstein constraint and gauge conditions, the solutions to the elliptic-hyperbolic system defined by the gauge fixed Einstein evolution equations are shown to give vacuum space-times.
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