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Open Access Publications from the University of California

An Analysis of the Conformal Formulation of the Einstein Constraint Equations on Asymptotically FlatManifolds

  • Author(s): Behzadan, Ali
  • et al.

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

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