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 (Class Quantum Gravity 30:205011, 2013), and Holst et al. (Phys Rev Lett 100(16):161101, 2008, Commun Math Phys 288(2):547–613, 2009), and also on the work of Maxwell (J Hyperbolic Differ Eqs 2(2):521–546, 2005a, Commun Math Phys 253(3):561–583, 2005b, Math Res Lett 16(4):627–645, 2009) and Dain (Class Quantum Gravity 21(2):555–573, 2004), under reasonable assumptions on the data we prove existence of both near- and far-from-constant mean curvature (CMC) solutions for a class of Robin boundary conditions commonly used in the literature for modeling black holes, with a third existence result for 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 Dainand 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 for other cases.
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