THE NEWTON POLYGON AND ELLIPTIC PROBLEMS WITH PARAMETER

In the study of the resolvent of a scalar elliptic operator, say, on a manifold without boundary there is a well-known Agmon-Agranovich-Vish ik condition of ellipticity with parameter which guarantees the existe nce of a ray of minimal growth of the resolvent. The paper is devoted to the investigation of the same problem in the case of systems which are elliptic in the sense of Douglis-Nirenberg. We look for algebraic conditions on the symbol providing the existence of the resolvent set containing a ray on the complex plane. We approach the problem using t he Newton polyhedron method. The idea of the method is to study simult aneously all the quasihomogeneous parts of the system obtained by assi gning to the spectral parameter various weights, defined by the corres ponding Newton polygon. On this way several equivalent necessary and s ufficient conditions on the symbol of the system guaranteeing the exis tence and sharp estimates for the resolvent are found. One of the equi valent conditions can be formulated in the following form: all the upp er left miners of the symbol satisfy ellipticity conditions. This subc lass of systems elliptic in the sense of Douglis-Nirenberg was introdu ced by A. KOZHEVNIKOV [K2].