Estimates for solutions of a parameter-elliptic multi-order system of differential equations

Abstract

This paper is concerned with a boundary value problem defined over a bounded region of Euclidean space, and in particular it is devoted to the establishment of a priori estimates for solutions of a parameter-elliptic multi-order system of differential equations under limited smoothness assumptions. In this endeavour we extend the results of Agranovich, Denk, and Faierman pertaining to a priori estimates for solutions associated with a parameter-elliptic scalar problem, as well as the results of various other authors who have extended the results of Agranovich et. al. from the scalar case to parameter-elliptic systems of operators which are either of homogeneous type or have the property that the diagonal operators are all of the same order. In addition, we extend some results of Kozhevnikov and Denk and Volevich who have also dealt with sytems of the kind under consideration here, in that one of the works of Kozhevnikov deals only with 2 × 2 systems, while the other, as well as the work of the last two authors, do not cover Dirichlet boundary conditions.