Mixed initial-boundary value problems for hyperbolic equations with constant coefficients

Abstract

We shall investigate the well-posedness and regularity of mixed initialboundary value problems for constant-coefficient strictly hyperbolic equations in a quarter space (MIBVP). The interface is a smooth compact manifold which divides the flat boundary into two regions. In this paper existence and uniqueness theorems are proved for a large class of such problems for hyperbolic equations of arbitrary order. These are based on the study of a new class of pseudo-differential operators ($.o.) which could naturally be called hyperbolic pseudo-d&f erential operators. A second paper will contain results on regularity for this class of problems, and it will give an asymptotic expansion of the trace of the solution near the interface. An example of a MIBVP is the wave equation with mixed DirichletNeumann boundary conditions. In three spatial dimensions it is given by the equations