  In this article we initiate a systematic study of the well-posedness theory
of the Einstein constraint equations on compact manifolds with boundary. This
is an important problem in general relativity, and it is particularly important
in numerical relativity, as it arises in models of Cauchy surfaces containing
asymptotically flat ends and/or trapped surfaces. Moreover, a number of
technical obstacles that appear when developing the solution theory for open,
asymptotically Euclidean manifolds have analogues on compact manifolds with
boundary. As a first step, here we restrict ourselves to the Lichnerowicz
equation, also called the Hamiltonian constraint equation, which is the main
source of nonlinearity in the constraint system. The focus is on low regularity
data and on the interaction between different types of boundary conditions,
which has not been carefully analyzed before. In order to develop a
well-posedness theory that mirrors the existing theory for the case of closed
manifolds, we first generalize the Yamabe classification to nonsmooth metrics
on compact manifolds with boundary. We then extend a result on conformal
invariance to manifolds with boundary, and prove a uniqueness theorem. Finally,
by using the method of sub- and super-solutions (order-preserving map
iteration), we establish several existence results for a large class of
problems covering a broad parameter regime, which includes most of the cases
relevant in practice.
