An Inverse Eigenvalue Problem and an Extremal Eigenvalue Problem

  • J. R. McLaughlin
Conference paper
Part of the Inverse Problems and Theoretical Imaging book series (IPTI)

Abstract

This talk presents results for two inverse problems which arise in the study of vibrating systems. The first problem (Part I) extends the theory of second order inverse eigenvalue problems in one dimension and is joint work with Carol Coleman. The second problem (Part II) solves an identification problem for composite membranes in n-dimensions; this work is joint work with Steven Cox.

Keywords

Dmax Carol 

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References

  1. [1]
    L. Andersson, Inverse problems with discontinuous coefficients, Inverse Problems 4 (1988) 353–397.CrossRefMATHADSMathSciNetGoogle Scholar
  2. [2]
    L. Andersson, Inverse eigenvalue problems for a Sturm-Liouville equation in impedance form, Inverse Problems (4) (1988) 929–971.Google Scholar
  3. [3]
    F. Atkinson, Discrete and Continuous Boundary Problems, Academic Press, New York, 1974.Google Scholar
  4. [4]
    G.Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math 78 (1946) 1–96.CrossRefMathSciNetGoogle Scholar
  5. [5]
    C. Coleman, An Inverse Spectral Problem with a Rough Coefficient, Ph.D. thesis, Rensselaer Polytechnic Institute, Troy, New York.Google Scholar
  6. [6]
    I. M. Gel’fand and B.M. Levitan, On the determination of a differential equation from its spectral function, Izv. Akad. Nauk SSSR Ser. Mat. 15 (1951) 309–360; English transl. Amer. Math. Soc. Trans1., 1 (1955) 253–304.MATHMathSciNetGoogle Scholar
  7. [7]
    J. R. McLaughlin, Analytical methods for recovering coefficients in differential equations from spectral data, SIAM Review 28 (1986) 53–72.CrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    J. Poeschel and E. Trubowitz, Inverse Spectral Theory, Academic Press, 1987.Google Scholar
  9. [9]
    H. Attouch, Variational Convergence for Functions and Operators, Pitman, Boston, 1984.MATHGoogle Scholar
  10. [10]
    G. Auchmuty, Dual Variational Principles for Eigenvalue Problems, in Nonlinear Functional Analysis and its Applications, F. Browder, ed., AMS 1986, 55–71.Google Scholar
  11. [11]
    V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, Reidel, Boston, 1986.MATHGoogle Scholar
  12. [12]
    J. Cea and K. Malanowski, An example of a max-min problem in partial differential equations, SIAM J Control, (3) 305–316.Google Scholar
  13. [13]
    S.J. Cox and J.R. McLaughlin, Extremal Eigenvalue Problems for Composite Membranes I, Appl Math and Optimization (to appear).Google Scholar
  14. [14]
    S.J. Cox and J.R. McLaughlin, Extremal Eigenvalue Problems for Composite Membranes II, Appl Math and Optimization (to appear).Google Scholar
  15. [15]
    B. Gidas, W. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979) 209–243.CrossRefMATHADSMathSciNetGoogle Scholar
  16. [16]
    B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Springer-Verlag, New York, 1985.MATHGoogle Scholar
  17. [17]
    M.G. Krein, On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability, AMS Translations Ser. 2 (1), (1955) 163–187.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • J. R. McLaughlin
    • 1
  1. 1.Department of Mathematical SciencesRensselaer Polytechnic InstituteTroyUSA

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