Singular Perturbation of an Improperly Posed Cauchy Problem

Abstract

In this paper we consider the solution of an improperly posed Cauchy problem (assumed to exist) for a coupled system of two second order elliptic differential equations one of which has a small coefficient $\varepsilon $ multiplying the highest order derivative. We compare the solution of this problem with the solution of the appropriately defined Cauchy problem for the elliptic differential equation resulting from setting $\varepsilon $ equal to zero. We prove that if the two solutions belong to the appropriate spaces of functions, then their difference in the $\mathcal{L}^2$-norm over some appropriately defined subdomain is of order $\varepsilon $ to some positive power which depends on the coefficients of the system and the particular subdomain considered.