  In this article we continue our effort to do a systematic development of the
solution theory for conformal formulations of the Einstein constraint equations
on compact manifolds with boundary. By building in a natural way on our recent
work in Holst and Tsogtgerel (2013), and Holst, Nagy, and Tsogtgerel (2008,
2009), and also on the work of Maxwell (2004, 2005, 2009) and Dain (2004),
under reasonable assumptions on the data we prove existence of both near- and
far-from-constant mean curvature solutions for a class of Robin boundary
conditions commonly used in the literature for modeling black holes, with a
third existence result for constant mean curvature (CMC) appearing as a special
case. Dain and Maxwell addressed initial data engineering for space-times that
evolve to contain black holes, determining solutions to the conformal
formulation on an asymptotically Euclidean manifold in the CMC setting, with
interior boundary conditions representing excised interior black hole regions.
Holst and Tsogtgerel compiled the interior boundary results covered by Dain and
Maxwell, and then developed general interior conditions to model the apparent
horizon boundary conditions of Dain and Maxwell for compact manifolds with
boundary, and subsequently proved existence of solutions to the Lichnerowicz
equation on compact manifolds with such boundary conditions. This paper picks
up where Holst and Tsogtgerel left off, addressing the general non-CMC case for
compact manifolds with boundary. As in our previous articles, our focus here is
again on low regularity data and on the interaction between different types of
boundary conditions. While our work here serves primarily to extend the
solution theory for the compact with boundary case, we also develop several
technical tools that have potential for use with the asymptotically Euclidean
case.
