# zbMATH — the first resource for mathematics

Characteristic initial value problem for hyperbolic systems of second order differential equations. (English) Zbl 0739.35039
A quasilinear hyperbolic system of second order differential equations is considered: $\sum_{a,b}g^{ab}{\partial^ 2u^ A\over \partial x^ a\partial x^ b}+\sum_{a,B}b^ a_ B{\partial u^ B\over \partial x^ a}+\sum_ Ba^ A_ Bu^ B=f^ A, (1)$ on $$L_ T\subset\mathbb{R}^{n+1}$$, $$L_ T$$ compact, $$(a,b=1,\ldots,n+1; A,B=1,\ldots,N)$$. (1) is considered for the unknown $$u:=(u^ A):=(u^ 1,\ldots,u^ N)$$. For example Einstein’s vacuum field equation (in harmonic coordinates) are of this type.
Initial data are given on two intersecting null (i.e. characteristic) hypersurfaces $$G^ j$$: (2) $$u^ A=u^ A_ j$$ on $$G^ j$$, $$j=1,2$$. The set $$G^ 1\cap G^ 2$$ is a spacelike $$(n-1)$$-dimensional surface; furthermore $$G^ 1\cup G^ 2$$ is part of the boundary of $$L_ T$$.
At first an existence theorem for the corresponding linear case is proven.
In order to obtain an $$s$$-times differentiable solution (in the sense of Sobolev spaces) one has to assume that the data are $$(2s-1)$$-times differentiable (with sufficiently large $$s$$) and that certain assumptions on the coefficients hold. This means that there is a gap of differentiability orders between the solution ($$s$$-times differentiable) and the data (($$2s-1$$)-times differentiable). It will be proven, that — in the generic case — this gap cannot be reduced by more than one half of differentiability order (in the sense of Sobolev spaces of fractional orders of differentiability).