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

  • R. DenkEmail author
  • M. Faierman
Conference paper
Part of the Operator Theory: Advances and Applications book series (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.

Keywords

Parameter-ellipticity multi-order systems a priori estimates 

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References

  1. ADN.
    S. Agmon, R. Douglis, and L. Nirenberg: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Comm. Pure Appl. Math. 17 (1964), 35-92.MathSciNetzbMATHCrossRefGoogle Scholar
  2. ADF.
    M.S. Agranovich, R. Denk, and M. Faierman: Weakly smooth nonselfadjoint elliptic boundary problems. In: Advances in partial differential equations: Spectral theory, microlocal analysis, singular manifolds, Math. Top. 14 (1997), 138-199.MathSciNetGoogle Scholar
  3. AV.
    M.S. Agranovich and M.I. Vishik: Elliptic problems with a parameter and parabolic problems of general form. Russ. Math. Surveys 19 (1964), 53-157.zbMATHCrossRefGoogle Scholar
  4. DHP.
    R. Denk, M. Hieber, and J. Pruss: Optimal Lp-Lq-regularity for parabolic problems with inhomogeneous boundary data. Math. Z. 257 (2007), 193-224.MathSciNetzbMATHCrossRefGoogle Scholar
  5. DF.
    R. Denk and M. Faierman: Estimates for solutions of a parameter-elliptic multi-order system of differential equations. Integral Equations Operator Theory 66 (2010), 327-365.MathSciNetzbMATHCrossRefGoogle Scholar
  6. DFM.
    R. Denk, M. Faierman, and M. Möller: An elliptic boundary problem for a system involving a discontinuous weight. Manuscripta Math. 108 (2001), 289-317.CrossRefGoogle Scholar
  7. DMV.
    R. Denk, R. Mennicken, and L. Volevich: The Newton polygon and elliptic problems with parameter. Math. Nachr. 192 (1998), 125-157.MathSciNetzbMATHCrossRefGoogle Scholar
  8. DV.
    R. Denk and L. Volevich: Elliptic boundary value problems with large parameter for mixed order systems. Amer. Math. Soc. Transl. (2) 206 (2002), 29-64.Google Scholar
  9. F.
    M. Faierman: Eigenvalue asymptotics for a boundary problem involving an elliptic system. Math. Nachr. 279 (2006), 1159-1184.MathSciNetzbMATHCrossRefGoogle Scholar
  10. G.
    G. Grubb: Functional calculus of pseudodifferential boundary problems 2nd edn. Birkhäuser, Boston, 1996.zbMATHCrossRefGoogle Scholar
  11. K1.
    A.N. Kozhevnikov: Spectral problems for pseudodifferential systems elliptic in the Douglis-Nirenberg sense, and their applications. Math. USSR Sobornik 21 (1973), 63-90.CrossRefGoogle Scholar
  12. K2.
    A.N. Kozhevnikov: Parameter-ellipticity for mixed order systems in the sense of Petrovskii. Commun. Appl. Anal. 5 (2001), 277-291.MathSciNetzbMATHGoogle Scholar
  13. T.
    H. Triebel: Interpolation theory, function spaces, differential operators. North-Holland, Amsterdam, 1978.Google Scholar
  14. VL.
    M.I. Vishik and L.A. Lyusternik: Regular degeneration and boundary layer for linear differential equations with small parameter. Amer. Math. Soc. Transl. (2) 20 (1962), 239-264.MathSciNetGoogle Scholar
  15. V.
    L.R. Volevich: Solvability of boundary value problems for general elliptic systems. Amer. Math. Soc. Transl. (2), 67 (1968), 182-225.zbMATHGoogle Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of KonstanzKonstanzGermany
  2. 2.School of Mathematics and StatisticsThe University of New South Wales UNSWSydneyAustralia

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