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We discuss the initial-boundary value problem of General Relativity. Previousconsiderations for a toy model problem in electrodynamics motivate theintroduction of a variational principle for the lapse with several attractiveproperties. In particular, it is argued that the resulting elliptic gaugecondition for the lapse together with a suitable condition for the shift andconstraint-preserving boundary conditions controlling the Weyl scalar Psi_0 areexpected to yield a well posed initial-boundary value problem for metricformulations of Einstein's field equations which are commonly used in numericalrelativity.  To present a simple and explicit example we consider the 3+1 decompositionintroduced by York of the field equations on a cubic domain with two periodicdirections and prove in the weak field limit that our gauge condition for thelapse and our boundary conditions lead to a well posed problem. The methoddiscussed here is quite general and should also yield well posed problems fordifferent ways of writing the evolution equations, including first ordersymmetric hyperbolic or mixed first-order second-order formulations. Well posedinitial-boundary value formulations for the linearization about arbitrarystationary configurations will be presented elsewhere.Comment: 34 pages, no figure

A minimization problem for the lapse and the initial-boundary value problem for Einstein's field equations