Estimates for solutions of reduced hyperbolic equations of the second order with a large parameter

We consider solutions of inhomogeneous, reduced hyperbolic equations of the second order, with a large parameter multiplying the unknown function. These solutions are defined on the m-dimensional region outside a star-shaped body. They satisfy an “outgoing” radiation condition at infinity and a Dirichlet boundary condition.We obtain a priori estimates for these solutions, at every point outside or on the surface of a two- or three-dimensional star-shaped body, that hold for sufficiently large values of the parameter. We prove that each solution is bounded by a linear combination of (i) the maximum norm of its prescribed boundary values, (ii) the L₂ norm of the prescribed values of its tangential derivative, (iii) an L₂ norm of the source term. This result is based on similar inequalities that we first obtain for a certain L₂ norm of the gradient, and of the normal derivative on the boundary, of each solution defined outside an m-dimensional star-shaped body.For the special case of the reduced wave equation, Morawetz and Ludwig 1] have obtained similar estimates. Just as their results have been used in 3] to confirm the geometrical theory of diffraction, the estimates obtained in this paper can be used to establish the validity of certain formal asymptotic solutions of reduced hyperbolic equations.