The Newton polygon and elliptic problems with parameter

Denk, Robert; Mennicken, Reinhard; Volevič, Leonid R.

The Newton polygon and elliptic problems with parameter

Dateien

The_Newton_polygon_and_elliptic_problems_with_parameter.pdf(362.87 KB)

Datum

1998

Autor:innen

Denk, Robert

Mennicken, Reinhard

Volevič, Leonid R.

Publikationstyp

Zeitschriftenartikel

Erschienen in

Mathematische Nachrichten ; 192 (1998). - S. 125-157

Zusammenfassung

In the study of the resolvent of a scalar elliptic operator, say, on a manifold without boundary there is a well-known Agmon-Agranovich-Vishik condition of ellipticity with parameter which guarantees the existence 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 the Newton polyhedron method. The idea of the method is to study simultaneously all the quasihomogeneous parts of the system obtained by assigning to the spectral parameter various weights, defined by the corresponding Newton polygon. On this way several equivalent necessary and sufficient conditions on the symbol of the system guaranteeing the existence and sharp estimates for the resolvent are found. One of the equivalent conditions can be formulated in the following form: all the upper left minors of the symbol satisfy ellipticity conditions. This subclass of systems elliptic in the sense of Douglis-Nirenberg was introduced by A. Kozhevnikov.

Fachgebiet (DDC)

510 Mathematik

Schlagwörter

Newton polygon,systems elliptic in the sense of Douglis Nirenberg,systems elliptic with parameter

Zitieren

ISO 690

DENK, Robert, Reinhard MENNICKEN, Leonid R. VOLEVIČ, 1998. The Newton polygon and elliptic problems with parameter. In: Mathematische Nachrichten. 192, pp. 125-157. Available under: doi: 10.1002/mana.19981920108

BibTex

@article{Denk1998Newto-719,
  year={1998},
  doi={10.1002/mana.19981920108},
  title={The Newton polygon and elliptic problems with parameter},
  volume={192},
  journal={Mathematische Nachrichten},
  pages={125--157},
  author={Denk, Robert and Mennicken, Reinhard and Volevič, Leonid R.}
}

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