The Characteristic Initial Value Problem for the Einstein-Yang-Mills-Higgs System in Weighted Sobolev Spaces

We revisit and complete existence and uniqueness results stated and partially established by Muller zum Hagen in 1990 for the characteristic initial value problem for quasilinear hyperbolic systems of second order with data prescribed on two intersecting smooth null hypersurfaces. The new ingredient of this investigation consists of some Moser estimates expressed in the same weighted Sobolev spaces as those used by Muller zum Hagen. These estimates, combined with energy inequalities obtained by Muller zum Hagen for the linearized Goursat problem, permit us to develop a fixed point method which leads clearly to an existence and uniqueness result for the quasilinear Goursat problem. As an application we locally solve, under finite differentiability conditions, the characteristic initial value problem for the Einstein-Yang-Mills-Higgs system using harmonic gauge for the gravitational potentials and Lorentz gauge for the Yang-Mills potentials.