Proof of uniqueness and membership in W 2 1 of the classical solution of a mixed problem for a self-adjoint hyperbolic equation

Abstract

In this article a uniqueness theorem for the classical solution of a mixed problem is proved under minimal assumptions on the coefficients of the differential operator for admitting the Fourier method of a hyperbolic second-order equation in an (N+1)-dimensional cylinder, whose cross section is a completely arbitrary bounded N-dimensional domain. Furthermore, it is proved that the classical solution of the indicated mixed problem, whenever it exists, belongs to the class W21 and is the generalized solution from W21 of the same problem.