Solutions of second order hyperbolic differential equations with constant coefficients in a domain with a plane boundary

Abstract

It is known that solutions of partial differential equations often have special regularity properties that set them apart from the “arbitrary” functions of any class C m or even C ∞. For example solutions of analytic elliptic equations are themselves analytic. Similarly, solutions of the heat equation are analytic in the space variables and of class C ∞ in the time. Even solutions of hyperbolic or ultra-hyperbolic equations have their special regularity properties, where however “regularity” does not always consist in possessing a large number of derivatives. This may find its expression in the fact that local integral transforms of the solution have many derivatives or even are analytic. Often the solutions form a family of functions, which share with the analytic functions the property of unique continuation, at least within certain limits.