On characteristic initial-value and mixed problems

Existence and uniqueness are proved for certain initial-value problems for hyperbolic systems of second-order differential equations, each having the same principal partg ^ab _{a b} (whereg ^ab is indefinite). The initial data are given on two intersecting hypersurfaces H₁ andH ₂ one of which-sayH ₁-is a characteristic surface. The other surface,H ₂, is permitted to be spacelike, timelike, or characteristic. For Einstein's vacuum field equations we restrict ourselves to anH ₂ that is characteristic. Unlike the Cauchy problem, the data have to be necessarily of a considerably higher differentiability class (Sobolev classW ^2m–1) than the solution (Sobolev classW ^m ). On the other hand, in the mixed problem (where one of the surfaces is spacelike) corner conditions have to be fulfilled. The occurrence of constraint equations for Einstein's metric field and for harmonic coordinates can be prevented by solving certain ordinary differential propagation equations.