result ([@foures-bruhat]) and its basic principles are now considered as well understood. In contrast, the systematic and general study of the initial boundary value problem for Einstein’s field equations started only recently and there are still basic open problems. There are two reasons for this.

The PDE theory of initial boundary value problems is technically more involved than that of the Cauchy problem and it has been developed in sufficient generality only in the last few decades (cf. [@benzoni-gavage:serre], [@kreiss:lorenz] for references). On the other hand, there seem to be only a few applications in general relativity whose geometry asks naturally for an analysis in terms of initial boundary value problems. Examples for this are the solutions of Anti-de Sitter type. They can be studied by prescribing for Einstein’s vacuum field equations with negative cosmological constant data at space-like and null infinity, which is represented in terms of the conformally rescaled and extended space-time by a time-like hypersurface ${\cal J}$, and by prescribing Cauchy data on a space-like initial hypersurface ${\cal S}$ which extends to ${\cal J}$. The study of this problem in [@friedrich:AdS] gives the first analysis of an initial boundary value problem for Einstein’s field equations which is 