On the Correct Formulation of a Multidimensional Problem for Strictly Hyperbolic Equations of Higher Order

A theorem of the unique solvability of the first boundary value problem in the Sobolev weighted spaces is proved for higherorder strictly hyperbolic systems in the conic domain with special orientation. In the space Rn, n > 2, let us consider a strictly hyperbolic equation of the form p(x, ∂)u(x) = f(x), (1) where ∂ = (∂1, . . . , ∂n), ∂j = ∂/∂xj , p(x, ξ) is a real polynomial of order 2m, m > 1, with respect to ξ = (ξ1, . . . , ξn), f is the known function and u is the unknown function. It is assumed that in equation (1) the coefficients at higher derivatives are constant and the other coefficients are finite and infinitely differentiable in Rn. Let D be a conic domain in Rn, i.e., D together with a point x ∈ D contains the entire beam tx, 0 < t < ∞. Denote by Γ the cone ∂D. It is assumed that D is homeomorphic onto the conic domain x1 + · · ·+ x2 n−1 − xn < 0, xn > 0 and Γ ′ = Γ\O is a connected (n− 1)-dimensional manifold of the class C∞, where O is the vertex of the cone Γ. Consider the problem: Find in the domain D the solution u(x) of equation (1) by the boundary conditions ∂iu ∂νi ∣