Propagation of Singularities of Fundamental Solutions of Hyperbolic Mixed Problems

Abstract

In this paper we shall deal with hyperbolic mixed problems with constant coefficients in a quarter-space and study the wave front sets of the fundamental solutions under the only assumption that the hyperbolic mixed problems are S-well posed. Recently Garnir has studied the wave front sets of fundamental solutions for hyperbolic systems [2]. The author was stimulated by his work. For the detailed literatures we refer the reader to [7], [8]. Now let us state our problems, assumptions and main results. Let R denote the ^-dimensional euclidean space and write x = (xl9 •••,o:n_1) for the coordinate x= (xl9 • • • , xn) in R n and £' — ($19 • • • , fn_a) , j = (f, £n+1) for the dual coordinate $ = (gl9 • • • , f n) . We shall also denote by JS+ the half-space {x = (x' , x^) e R; xn^>0}. For differentiation we will use the symbol D = i~ (d/dxl9 •-, d/dxn) . Let P-P(f) be a hyperbolic polynomial of order m of n variables f with respect to t?= (1,0, • • • , 0) in the sense of Carding, i.e.