UC San Diego

In this thesis we consider the conformal formulation of the Einstein constraintequations on asymptotically flat (AF) manifolds. The conformal method transforms the original underdetermined system of constraint equations into a potentially well-posed nonlinear elliptic system which is referred to as Lichnerowicz-Choquet-Bruhat-York (LCBY) system. We investigate the important properties of weighted Sobolev spaces as the appropriate solution spaces for the LCBY equations on AF manifolds. We combine elliptic estimates, sub- and supersolution constructions, fixed-point theorems, and Fredholm-Riesz-Schauder theory to establish existence of non-CMC weak solutions of the LCBY equations for AF manifolds of class Ŵ{s,p}_{\delta} where p\in (1,\infty), s \in (1+3/p ,\infty), -1<\delta<0, with metric in the positive Yamabe class