Mixed Problem for Hyperbolic Systems of First Order

Abstract

In the last ten years, a general theory of mixed problems for linear hyperbolic systems of first order has been developed. In this paper, we treat the L2 well-posed mixed problems for strongly hyperbolic systems of first order with constant coefficients. It is well known that Cauchy problem for hyperbolic systems of first order with constant coefficients is L2 well-posed if and only if it is strongly hyperbolic DL2j. The family of strongly hyperbolic systems contains the strictly and symmetric hyperbolic systems. But, Strang's condition for strong hyper bolicity seems to be useless except for Cauchy problem. So, in Sec. 3, we prove another condition for strong hyperbolicity which is useful for not only Cauchy problem, but also mixed problems, the lacunas of Riemann's matrix of strongly hyperbolic systems [1], the propagation of singularities Q8] and others. The L2 well-posed mixed problems for the strictly and symmetric hyperbolic systems have been fully investigated. But, even if hyperbolic systems have constant coefficients, nothing is known in general for ones with multiple characteristics except for systems with constant multiple characteristics [[KT] and 2x2 systems [JL3]. We consider the mixed problem for hyperbolic systems of first order with constant coefficients: