  We discuss the initial-boundary value problem of General Relativity. Previous
considerations for a toy model problem in electrodynamics motivate the
introduction of a variational principle for the lapse with several attractive
properties. In particular, it is argued that the resulting elliptic gauge
condition for the lapse together with a suitable condition for the shift and
constraint-preserving boundary conditions controlling the Weyl scalar Psi_0 are
expected to yield a well posed initial-boundary value problem for metric
formulations of Einstein's field equations which are commonly used in numerical
relativity.
  To present a simple and explicit example we consider the 3+1 decomposition
introduced by York of the field equations on a cubic domain with two periodic
directions and prove in the weak field limit that our gauge condition for the
lapse and our boundary conditions lead to a well posed problem. The method
discussed here is quite general and should also yield well posed problems for
different ways of writing the evolution equations, including first order
symmetric hyperbolic or mixed first-order second-order formulations. Well posed
initial-boundary value formulations for the linearization about arbitrary
stationary configurations will be presented elsewhere.
