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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).
Reviewer: I.Badea (Craiova)

MSC:
35L55 Higher-order hyperbolic systems
35L15 Initial value problems for second-order hyperbolic equations
35L85 Unilateral problems for linear hyperbolic equations and variational inequalities with linear hyperbolic operators
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