  We consider the conformal decomposition of Einstein's constraint equations
introduced by Lichnerowicz and York, on a compact manifold with boundary. We
use order relations on appropriate Banach spaces to derive weak solution
generalizations of known sub- and super-solutions (barriers) for the
Hamiltonian constraint. The barriers are combined with variational methods to
establish existence of solutions to the Hamiltonian constraint in the
intersection of the space of essentially bounded functions and the Sobolev
space H1. The result is established for scalar curvature R of the background
metric having any sign; non-negative R requires additional positivity
assumptions either on the matter energy density or on the trace-free
divergence-free part of the extrinsic curvature. Although the formulation is
different, the result can be viewed as lowering the regularity of the recent
result of Maxwell on rough CMC solutions. We also establish existence of
non-CMC solutions of the Hamiltonian and momentum constraint equations. The
result is obtained using fixed-point iteration and compactness arguments
directly, rather than by building a contraction map. The non-CMC result can be
viewed as a type of extension of the regularity of the 1996 non-CMC result of
Isenberg and Moncrief to lower regularity and to scalar curvature R having any
sign.
