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Transition Layers, Angular Limiting Solutions and Internal Layers in Singularly Perturbed Nonlinear Eigenvalue Problems

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Singular Perturbations in Systems and Control

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 280))

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Abstract

In this review article we shall survey recent developments in singular perturbations of nonlinear eigenvalue problems (N.L.E.P.). Nonlinear eigenvalue problems in bounded domains characteristically exhibit multiple solutions; as the eigenvalue parameter varies, connected components of solutions go through bending points (turning points), which mathematically are closely related to bifurcations. Singularly perturbed nonlinear eigenvalue problems exhibit a wealth of singular limits; including boundary layers, angular limiting solutions, transition layers, internal layers, and free boundary limits. In the first sections after reviewing basic facts for N.L.E.P.’s, we will survey the asymptotic behavior of branches of solutions as a nonlinear eigenvalue parameter goes to infinity. In the second section, we will emphasize the close feedback and coupling between the generic multiplicity of solutions of N.L.E.P.’s and the corresponding multiplicity of singular limits. Typically, the singular limit problems are free boundary problems (F.B.P.) with internal layers centered on the a priori unknown free boundaries; moreover, the free boundaries themselves can exhibit multiplicity, with their bending points (bifurcations) not necessarily related to bending points of the original perturbed N.L.E.P. The plan of the article is as follows:

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Author information

Authors and Affiliations

  1. Departement de Mathematiques — Informatique — Systemes, Ecole Centrale de Lyon, 69130, Ecully, France

    Claude M. Brauner (Consultant)

  2. Los Alamos National Laboratory, USA

    Claude M. Brauner (Consultant)

  3. Center for Nonlinear Studies, MS B258, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    Basil Nicolaenko

Authors
  1. Claude M. Brauner
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  2. Basil Nicolaenko
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Editors and Affiliations

  1. M.S. 210-9, NASA-Ames Research Center, 94035, Moffett Field, CA, USA

    M. D. Ardema

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© 1983 Springer-Verlag Wien

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Brauner, C.M., Nicolaenko, B. (1983). Transition Layers, Angular Limiting Solutions and Internal Layers in Singularly Perturbed Nonlinear Eigenvalue Problems. In: Ardema, M.D. (eds) Singular Perturbations in Systems and Control. International Centre for Mechanical Sciences, vol 280. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2638-7_11

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  • DOI: https://doi.org/10.1007/978-3-7091-2638-7_11

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-81751-3

  • Online ISBN: 978-3-7091-2638-7

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