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Necessity of Parameter-ellipticity for Multi-order Systems of Differential Equations

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Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 221))

Abstract

In this paper we investigate parameter-ellipticity conditions for multi-order systems of differential equations on a bounded domain.Unde r suitable assumptions on smoothness and on the order structure of the system, it is shown that parameter-dependent a priori estimates imply the conditions of parameter-ellipticity, i.e., interior ellipticity, conditions of Shapiro- Lopatinskii type, and conditions of Vishik-Lyusternik type.T he mixed-order systems considered here are of general form; in particular, it is not assumed that the diagonal operators are of the same order.Th is paper is a continuation of an article by the same authors where the sufficiency was shown, i.e., a priori estimates for the solutions of parameter-elliptic multi-order systems were established.

Mathematics Subject Classification (2000). Primary 35J55; Secondary 35S15.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Konstanz, D-78457, Konstanz, Germany

    R. Denk

  2. School of Mathematics and Statistics, The University of New South Wales UNSW, Sydney, NSW, 2052, Australia

    M. Faierman

Authors
  1. R. Denk
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  2. M. Faierman
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Corresponding author

Correspondence to R. Denk .

Editor information

Editors and Affiliations

  1. , Abt. Mathematik V, Universität Ulm, Ulm, 89069, Germany

    Wolfgang Arendt

  2. Department of Mathematics, Virginia Polytechnic Institute, McBryde Hall 524, Blacksburg, 24061, Virginia, USA

    Joseph A. Ball

  3. Institut fĂĽr Mathematik, AG Modellierung, Numerik und, TU Berlin, StraĂźe des 17 Juni 136, Berlin, 10623, Berlin, Germany

    Jussi Behrndt

  4. Fak. II Mathematik & Naturwissenschaften, Institut fĂĽr Mathematik, TU Berlin, StraĂźe des 17. Juni 136, Berlin, 10623, Berlin, Germany

    Karl-Heinz Förster

  5. Fak. II Mathematik & Naturwissenschaften, Institut fĂĽr Mathematik, TU Berlin, StraĂźe des 17. Juni 136, Berlin, 10623, Berlin, Germany

    Volker Mehrmann

  6. , Institut fĂĽr Mathematik, TU Ilmenau, Weimarer Str. 25, Ilmenau, 98693, ThĂĽringen, Germany

    Carsten Trunk

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Denk, R., Faierman, M. (2012). Necessity of Parameter-ellipticity for Multi-order Systems of Differential Equations. In: Arendt, W., Ball, J., Behrndt, J., Förster, KH., Mehrmann, V., Trunk, C. (eds) Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations. Operator Theory: Advances and Applications, vol 221. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0297-0_14

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