Boundary Value and Mixed Problems for General Hyperbolic Systems

Abstract

Boundary value and mixed problems for hyperbolic equations have been studied in classes of sufficiently smooth functions by Sakamoto [Sak], Kreiss [Kre], Agranovich [Agr], Chazarin and Piriou [Volevich and Gindikin [yoGI, and others (see the survey [VIv]). In this chapter we study boundary value and mixed problems for systems strictly hyperbolic in the Leray— Volevich sense in complete scale of spaces of Sobolev type depending on parameters s, τ ∈ R; s characterizes the smoothness of the solutions in all the variables, and τ characterizes the additional smoothness with respect to the tangential variables. The smaller s and τ are, the more generalized the solution is; for suffuciently large s and τ the solution is the ordinary classical solution of the problem. The result obtained enable us, in particular, to investigate hyperbolic problems with arbitrary power singularities on the right hand sides, to construct and investigate the Green’s matrices of the problems under study, and to investigate the class of degenerate hyperbolic problems for systems of equations.