Mixed boundary value problems for nonstrict hyperbolic equations

Abstract

Let Q c ƒ?", n ^ 2, be a bounded open set with boundary dQ a regularly imbedded C 00 submanifold of R n . Let a(y; £, X) be the polynomial Let a 0 (y;^ X) be the principal part, the terms of total degree p. We assume that, for yeU^eR n 9 the roots of a 0 (y;^, •) are all real and nonzero. This implies p = 2r is even. Let b k (y\ Ç, X) be polynomials of order p k < p, k = 1,2, ...,r, with principal part b k0 (y; <!;, X\ yedQ £eC p , XeC. We assume the coefficients of these polynomials are C 00 . We also assume the appropriate version of the "root covering condition" familiar from elliptic boundary value problems. Specifically, for y e 3Q, let v = v(y) be the unit vector in R n in the direction of the inner normal to dCl at y, and let W = W(y) be the subspace of R n orthogonal to v. The assumptions imply that for I € Wand Re X > 0, or <J e W\(0) and X = 0, a 0 (y; /iv + i£, X) has precisely r roots /^(£, X\ ..., /*,.(£, X) with negative real part. Let q(n;£,X) = no*-lij&X)).AM S 1970 subject classifications. Primary 35L35.