On the Shapiro–Lopatinkii condition for elliptic problems

This paper considers elliptic problems with high-order derivatives multiplied by a small parameter. We found the algebraic conditions for an operator and the boundary conditions that guarantee the Fredholm property. An a priori estimate for the solution with a constant independent of the small parameter is proved. These results are known for elliptic boundary-value problems with small parameter in the half-space R _n ⁺. We extend them to the case of bounded domains with smooth boundary. The small-parameter coercive conditions are formulated, and a two-sided estimate is proved.