Regularité conormale classique des problèmes de Cauchy et de reflexion transverse pour un système 2\(\times 2\) semilinéaire. (Classical conormal regularity for Cauchy problems and transversal reflection for a two by two semilinear system) (Q914954)

A short note of this paper was published in C. R. Acad. Sci., Paris, Sér. I 309, No.13, 817-820 (1989) see Zbl 0702.35039 above. We consider a first order semilinear \(2\times 2\) system in an open set of \({\mathbb{R}}^ n\), a non-characteristic hypersurface S and a hypersurface \(\Gamma\) of S. We suppose that through \(\Gamma\), there are exactly two characteristic transversal hypersurfaces \(\Sigma_ 1,\Sigma_ 2\) and that the bicharacteristics on \(\Sigma_ 1,\Sigma_ 2\) are transversal to \(\Gamma\). We suppose that u is a solution in \(\Omega\), one of the two half-spaces delimited by S and that u is the restriction to \(\Omega\) of a piecewise conormal distribution relatively to \(\Sigma_ 1\cup \Sigma_ 2.For\) the Cauchy problem, we show that if the trace of u on S is classical relatively to \(\Gamma\), so u is classical relatively to \(\Sigma_ 1\) and \(\Sigma_ 2\). For the boundary value hyperbolic problem with boundary condition on S, satisfying uniform Lopatinski condition, we show that if the boundary condition is classical relatively to \(\Gamma\) and if u is classical relatively to \(\Sigma_ 1\), supposed outgoing, then u is classical relatively to the reflected hypersurface \(\Sigma_ 2\).