On the Solvability of a Spatial Problem of Darboux Type for the Wave Equation

The question of the correct formulation of one spatial problem of Darboux type for the wave equation has been investigated. The correct formulation of that problem in the Sobolev space has been proved for surfaces having a quite definite orientation on which are given the boundary value conditions of the problem of Darboux type. In the space of variables x1, x2, t we consider the wave equation ƒu ≡ ∂ 2u ∂t2 − ∂ 2u ∂x1 − ∂ 2u ∂x2 2 = F, (1) where F is the known and u is the unknown function. Denote by D+ : 0 < x2 < t, 0 < t < t0, the domain lying in a half-space t > 0 bounded by a time-type plane surface S0 : x2 = 0, 0 ≤ t ≤ t0, a characteristic surface S1 : t − x2 = 0, 0 ≤ t ≤ t0, of equation (1), and a plane t = t0. Consider the problem of Darboux type formulated as follows: find in the domain D+ the solution u(x1, x2, t) of equation (1) by the following boundary conditions: u ∣ ∣ S1 = f1 (2) and ∂u ∂n ∣