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Generalized solutions and non-uniqueness in the Einstein constraint equations : some unresolved issues with the conformal formulation


In this thesis we consider the problem of determining solutions to the conformal formulation of the Einstein constraint equations with low regularity coefficients and then discuss certain non-uniqueness properties of the conformal formulation of the constraints on a closed manifold. We first investigate the existence of a solution to a semi-linear, elliptic, partial differential equation with distributional coefficients and boundary conditions that models the problem that one encounters when studying the Einstein constraint equations with low regularity data and constant mean curvature. Our method for solving this problem consists of solving a net of regularized, semi- linear problems with data obtained by smoothing the original, distributional coefficients. We then obtain a net of solutions and show that this net satisfies certain decay estimates by determining estimates for sub- and super-solutions and utilizing classical, a priori elliptic estimates. The estimates for this net of solutions allow us to regard this collection of functions as a solution in a Colombeau-type algebra. Following our analysis of this low regularity problem, we consider whether or not solutions to the conformal formulation of the constraints with an unscaled matter source are unique. For positive, constant scalar curvature and constant mean curvature, we first demonstrate the existence of a critical energy density for the Hamiltonian constraint. We then show that for this choice of energy density, the linearization of the elliptic system develops a one-dimensional kernel in both the constant mean curvature and non-constant mean curvature cases. Using a Liapunov-Schmidt reduction and standard techniques from bifurcation theory, we demonstrate that solutions to the conformal formulation with an unscaled data source are non-unique by determining an explicit solution curve and analyzing its behavior in the neighborhood of a particular solution

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