Integral Approach to Sensitive Singular Perturbations



The main purpose of this chapter is to give general ideas on a kind of singular perturbation arising in thin shell theory when the middle surface is elliptic and the shell is fixed on a part of the boundary and free on the rest, as well as an integral heuristic procedure reducing these problems to simpler ones. The system depends essentially on the parameter ε equal to the relative thickness of the shell. It appears that the “limit problem” for ε = 0 is highly ill posed. Indeed, the boundary conditions on the free boundary are not “adapted” to the system of equations; they do not satisfy the Shapiro–Lopatinskii (SL) condition. Roughly speaking, this amounts to some kind of “transparency” of the boundary conditions, which allows some kind of locally indeterminate oscillations along the boundary, exponentially decreasing inside the domain. This pathological behavior only occurs for ε = 0. In fact, for ε > 0 the problem is “classical” When ε is positive but small, the “determinacy” of the oscillations only holds with the help of boundary conditions on other boundaries, as well as the small terms coming from ε > 0.


Integral Approach Singular Perturbation Inverse Fourier Transform Distribution Space Principal Part 


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  1. [AgDoNi59]
    Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Comm. Pure Appl. Math., 12, 623-727 (1959).MATHCrossRefMathSciNetGoogle Scholar
  2. [BeMiSa08]
    Bechet, F., Millet, O., Sanchez-Palencia, É.: Singular perturbations generating complexification phenomena for elliptic shells. Comput. Mech. 43 (2):207-221 (2008).CrossRefMathSciNetGoogle Scholar
  3. [CoHi62]
    Courant, R., Hilbert, D.: Methods of Mathematical Physics, Vol. II, Interscience, New York (1962).MATHGoogle Scholar
  4. [Ec79]
    Eckhaus, W.: Asymptotic Analysis of Singular Perturbations, North-Holland, Amsterdam (1979).MATHGoogle Scholar
  5. [EgMeSa07]
    Egorov, Y.V., Meunier, N., Sanchez-Palencia, É.: Rigorous and heuristic treatment of certain sensitive singular perturbations. J. Math. Pures Appl., 88, 123-147 (2007).MATHMathSciNetGoogle Scholar
  6. [EgMeSa09]
    Egorov, Y.V., Meunier, N., Sanchez-Palencia, É.: Rigorous and heuristic treatment of sensitive singular perturbations in shell theory, in Approximation Theory and Partial Differential Equations. Topics Around the Research of Vladimir Maz'ya (to appear).Google Scholar
  7. [EgSc97]
    Egorov, Y.V., Schulze, B.W.: Pseudo-differential Operators, Singularities and Applications, Birkhäuser, Berlin (1997).Google Scholar
  8. [GeCh64]
    Gelfand, I.M., Shilov, G.: Generalized Functions, Academic Press, New York and London (1964).Google Scholar
  9. [Il91]
    Il'in, A.M.: Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, American Mathematical Society, Providence, RI (1991).Google Scholar
  10. [MeSa06]
    Meunier, N., Sanchez-Palencia, É.: Sensitive versus classical perturbation problem via Fourier transform. Math. Models Methods Appl. Sci., 16, 1783-1816 (2006).MATHCrossRefMathSciNetGoogle Scholar
  11. [MeEtAl07]
    Meunier, N., Sanchez-Hubert, J., Sanchez Palencia, E.: Various kinds of sensitive singular perturbations. Ann. Math. Blaise Pascal, 14, 199-242 (2007).MATHMathSciNetGoogle Scholar
  12. [SaHuSa97]
    Sanchez-Hubert, J., Sanchez-Palencia, É.: Coques Élastiques Minces. Propriétés Asymptotiques, Masson, Paris (1997).MATHGoogle Scholar
  13. [Sc50]
    Schwartz, L.: Théorie des Distributions, Hermann, Paris (1950).MATHGoogle Scholar
  14. [Ta81]
    Taylor, M.E.: Pseudodifferential Operators, Princeton Univ. Press, Princeton, NJ (1981).MATHGoogle Scholar

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© Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Université de Paris DescartesParisFrance
  2. 2.Université Pierre et Marie Curie and CNRS, Institut Jean Le Rond D’AlembertParisFrance

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