In multidimension, the Cauchy problem for a first order quasilinear strictly
hyperbolic system with a continuous data whose first derivatives are discontinuous
on an hypersurface, is a free characteristic boundaries problem. We solve it locally
in a blown-up configuration into conormal Sobolev spaces: thanks to a
paralinearization and the good unknown Alinhac's idea, we can use an iterative
scheme without loss of derivatives.
Key words: gradients waves, quasilinear hyperbolic system, characteristic
surfaces, conormal spaces, paralinearization, tame inequality.