Boundary value problems for elliptic operator pencils

In this paper boundary value problems are studied for systems with large parameter, elliptic in the sense of Douglis–Nirenberg. We restrict ourselves on model problems acting in the half-space. It is possible to define parameter-ellipticity for such problems, in particular we formulate Shapiro–Lopatinskii type conditions on the boundary operators. It can be shown that parameter-elliptic boundary value problems are uniquely solvable and that their solutions satisfy uniform a priori estimates in parameter-dependent norms. We essentially use ideas from Newton's polygon method and of Vishik–Lyusternik boundary layer theory.