Rough solutions of the Einstein constraint equations with nonconstant mean curvature

Abstract

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.