Improperly Posed Initial Value Problems for Self-Adjoint Hyperbolic and Elliptic Equations

Abstract

Integral representations are obtained for the solution to Cauchy’s problem for hyperbolic equations along a convex time-like surface, the exterior characteristic initial value problem for hyperbolic equations, and Cauchy’s problem for elliptic equations along an analytic surface. Each of these problems is improperly posed in the real domain and hence our representations are constructed by integrating over appropriate regions in the space of one and several complex variables.