On Some Boundary Value Problems for an Ultrahyperbolic Equation

The correct formulation of a characteristic problem and a Darboux type problem in the special weighted functional spaces for an ultrahyperbolic equation is investigated. In the space of variables x1, x2, y1 and y2 we consider the ultrahyperbolic equation uy1y1 + uy2y2 − ux1x1 − ux2x2 = F. (1) Denote by D : −y1 < x1 < y1, 0 < y1 < +∞, a dihedral angle bounded by the characteristic surfaces S1 : x1 − y1 = 0, 0 ≤ y1 < +∞, and S2 : x1 + y1 = 0, 0 ≤ y1 < +∞, of equation (1). We shall consider a characteristic problem formulated as follows: in the domain D find a solution u(x1, x2, y1, y2) of equation (1) by the boundary conditions u |Si= fi, i = 1, 2, (2) where fi, i = 1, 2, are given real functions on Si and (f1 − f2) |S1∩S2= 0. Characteristic problems formulated similarly were considered in [1–3]. Let G = {(x1, ξ1, y1, ξ2) ∈ R4 : −y1 < x1 < y1, 0 < y1 < +∞; −∞ < ξi < +∞, i = 1, 2}. Denote by Φk(D), k ≥ 2, the space of functions u(x1, x2, y1, y2) of the class Ck(D) whose partial Fourier transforms û(x1, ξ1, y1, ξ2) with respect to the variables x2 and y2 are continuous functions in G together with partial derivatives with respect to the variables x1 and y1 up to kth order inclusive and satisfy the following estimates: for any natural N there exist positive 1991 Mathematics Subject Classification. 35L20.