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.

This paper originated in some stimulating conversations the author had with H. Lewy in the spring of 1953. Lewy’s work on elliptic equations has been of considerable influence on the treatment of the hyperbolic case presented here.