On elliptic operator pencils with general boundary conditions

Abstract

In this paper operator pencils $A(x,D,\lambda)$ are investigated which depend polynomially on the parameter $\lambda$ and act on a manifold with boundary. The operator A is assumed to satisfy the condition of N-ellipticity with parameter which is an ellipticity condition formulated with the use of the Newton polygon. We consider general boundary operators $B_1(x,D),...,B_m(x,D)$ and define N-ellipticity for the boundary value problem $(A,B_1,...,B_m)$ analogously to the Shapiro-Lopatinskii condition. It is shown that the boundary value problem is N-elliptic if and only if an a priori estimate holds, where the norms in the estimate are again defined in terms of the Newton polygon. These results are closely connected with singular perturbation theory and lead to uniform estimates for problems of Vishik-Lyusternik type containing a small parameter.