Annali di Matematica Pura ed Applicata

, Volume 148, Issue 1, pp 367–395 | Cite as

Coercive singular perturbations: Eigenvalue problems and bifurcation phenomena

  • L. S. Frank


The method based upon a constructive reduction of coercive singular perturbations to regular ones, introduced in 1977 (see [4]) and developed later on (see [9–11]) is applied for computing the asymptotic expansions for eigenvalues of coercive singular perturbations, when the small parameter goes to zero. The same method turns out to be useful for investigating the asymptotic behaviour of solutions to quasi-linear coercive singular perturbations in the neighbourhood of the bifurcation points. It can be applied to classes of quasi-linear singular perturbations whose principal linear part in local representation is coercive and the nonlinear part is analytic in some ball in the solution space with values in the data space. The results are summarized in [7, 8].


Asymptotic Behaviour Asymptotic Expansion Eigenvalue Problem Small Parameter Solution Space 
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  1. [1]
    M. Birman,The spectrum of singular boundary value problems, Math. Sb.,55 (1961), pp. 124–174; AMS Translation series,53 (1966), pp. 23–80.Google Scholar
  2. [2]
    L. Boutet de Monvel,Boundary problems for pseudo-differential operators, Acta Math.,126, 1–2 (1971), pp. 11–51.Google Scholar
  3. [3]
    L. S. Frank,General boundary value problems for ordinary differential equations with small parameter, Ann. Mat. Pura Appl. (IV),114 (1977), pp. 27–67.Google Scholar
  4. [4]
    L. S. Frank,Perturbazioni singolari ellittiche, Rend. Milano,47 (1977), pp. 135–163.Google Scholar
  5. [5]
    L. S. Frank,Coercive singular perturbations I: A priori estimates, Ann. Mat. Pura Appl. (IV),119 (1979), pp. 41–113.Google Scholar
  6. [6]
    L. S. Frank,L'inégalité de Garding et perturbations singuliéres elliptiques aux différences finies, C. R. Acad. Sc., Paris. Série I, t.296 (1983), pp. 93–96.Google Scholar
  7. [7]
    L. S. Frank,Perturbations singuliéres coercives IV: Problème des valeurs propres, C. R. Acad. Sci., Paris, Séne I, t.301, n. 3 (1985), pp. 69–72.Google Scholar
  8. [8]
    L. S. Frank,Remarque sur le phénomène de bifurcation pour les perturbations singulières coercives quasi-linéaires, C. R. Acad. Sci., Paris, Série I, t.302, n. 1 (1986), pp. 17–20.Google Scholar
  9. [9]
    L. S. Frank andW. D. Wendt,Coercive singular perturbations: asymptotics and reduction to regularly perturbed boundary value problems, in:Analytical and Numerical approaches to asymptotic problems in Analysis (O. Axelsson,L. S. Frank,A. van der Sluis Editors), North-Holland, Amsterdam (1981), pp. 305–318.Google Scholar
  10. [10]
    L. S. Frank -W. D. Wendt,Coercive singular perturbations II: Reduction to regular perturbations and applications, Comm. P.D.E.,7 (1982), pp. 469–535.Google Scholar
  11. [11]
    L. S. Frank -W. D. Wendt,Coercive singular perturbations III: Weiner-Hopf operators, J. d'Analyse Mathématique, vol.43 (1983/84), pp. 88–135.Google Scholar
  12. [12]
    W. M. Greenlee, Manuscripta Math.,43 (1981), pp. 157–174.Google Scholar
  13. [13]
    P. P. N.De Groen,Singular perturbations of spectra, report TW 185/78, Math. Centrum Amsterdam (1978), pp. 1–24.Google Scholar
  14. [14]
    T. Kato,Perturbation theory for linear operators, Springer-Verlag, Berlin, (1966).Google Scholar
  15. [15]
    M. I. Vishik -L. A. Lyusternik, Uspekhi Mat. Nauk,12 (1957), pp. 3–122.Google Scholar
  16. [16]
    W. D. Wendt,Coercive singularity perturbed Wiener-Hopf operators and applications, Ph.D. Thesis, Catholic University, Nijmegen (1983).Google Scholar
  17. [17]
    L. S.Frank and J. J.Heijstek, Report No. 8604 (1986), University of Nijmegen.Google Scholar

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© Fondazione Annali di Matematica Pura ed Aplicata 1987

Authors and Affiliations

  • L. S. Frank
    • 1
  1. 1.Mathematische InstitutKatholieke UniversiteitED NijmegenNetherlands

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