Interior Estimates for Singularly Perturbed Problems

Abstract

The solution of the Dirichlet problem for a singularly perturbed elliptic differential equation $ϵL_{1}u+L_{0}u=h$ of order $2m$ converges, for $ϵ\to 0$, outside of the boundary layer uniformly to a solution of the degenerate elliptic equation $L_{0}w=h$ of lower order. It is shown in the case of order zero of $L_0$ this assertion may be proved immediately, i.e., without the usual construction of boundary layer terms, but rather elementary and on weak smoothness conditions with respect to the boundary of the domain.