On the Shapiro-Lopatinkii condition for elliptic problem

Abstract

This paper is concerned with elliptic problems including a small parameter multiplying higher order derivatives. We found algebraic conditions on the operator and boundary conditions which guarantee the Fredholm property, and prove an a priori estimate for the solution with a constant independent of the small parameter. These results are known for elliptic boundary value problems with small parameter in the half space Rⁿ+. We extend them to the case of bounded domains with smooth boundary. The small parameter coercive conditions are formulated and two-sided estimate is proved.