M. Dossa and C. Tadmon (2010) “The Characteristic Initial Value Problem,” Applied Mathematics Research eXpress, Vol. 2010, No. 2, pp. 154–231 doi:10.1093/amrx/abq014 The Characteristic Initial Value Problem for the Einstein–Yang–Mills–Higgs System in Weighted Sobolev Spaces Marcel Dossa 1 and Calvin Tadmon 2 1 Department of Mathematics, Faculty of Science, University of Yaounde I, Po. Box 812, Yaounde, Cameroon and 2 Department of Mathematics and Computer Science, Faculty of Science, University of Dschang, Po. Box 67, Dschang, Cameroon Correspondence to be sent to: marceldossa@yahoo.fr We revisit and complete existence and uniqueness results stated and partially established by M ¨ uller zum Hagen in 1990 for the characteristic initial value problem for quasilinear hyperbolic systems of second order with data prescribed on two inter- secting 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 M¨ uller zum Hagen. These estimates, combined with energy inequalities obtained by M ¨ uller 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 quasi- linear 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. 1 Introduction This work is devoted to the characteristic initial value problem for the Einstein–Yang– Mills–Higgs (EYMH) system with initial data prescribed on two intersecting smooth null hypersurfaces. The interests and physical motivations for studying characteris- tic initial value problems have been widely mentioned in [34, 37]. These problems, for example, play a fundamental role in the recent theory of black holes formation made Received February 28, 2010; Revised June 10, 2010; Accepted July 22, 2010 c  The Author(s) 2010. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org. at BIUS Jussieu on September 13, 2010 amrx.oxfordjournals.org Downloaded from