On Weak Well-posedness of Mixed Problems for Hyperbolic Systems

Abstract

I, INTRODUCTION. In this paper, we treat a mixed problem for first order hyperbolic systems of partial differential equations with constant coefficients in quarter space and show a necessary and sufficient condition for the weak L-wellposedness of mixed problems. For equations hyperbolic in Garding's sense, the mixed problem has been investigated by Hersh ([1] and [2]), stating that the condition that the Lopatinski determinant of the problem is not equal to zero (in this paper this is represented in the form (3)) is equivalent to the weak Z,-wellposedness. But his proof is incomplete by the following reasons. First the generalized eigenvectors Wj are used in the construction of the s^. elementary solution C7(r) (in [2]; also in [1] we refer to z;), but these Wj were only constructed in pointwise sense, hence, if the multiplicity of eigenvalues changes, the construction must also be changed. Thus the smoothness of the construction of w} with respect to the variables r and ^ should have been shown. Secondly, to show that the elementary solution increases at most by polynomial order as ] r | 2 + |^|->-r°o, the explicit formula of the generalized eigenvectors wjy could not be used, since the formula itself depends on (r, ??). We shall here show that these difficulties can be overcome by using Seidenberg-Tarski elimination theorem and the Cauchy integral formula for matrices and vectors.