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On the structure of layers for singularly perturbed equations in the case of unbounded energy

Published online by Cambridge University Press:  15 August 2002

E. Sanchez–Palencia
E. Sanchez–Palencia*
Affiliation:
Laboratoire de Modélisation en Mécanique, CNRS-Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05, France; sanchez@lmm.jussieu.fr.
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Abstract

We consider singular perturbation variational problemsdepending on a small parameter ε. The right hand side is suchthat the energydoes not remain bounded as ε → 0. The asymptoticbehavior involves internallayers where most of the energy concentrates. Three examples are addressed,with limits elliptic, parabolic and hyperbolic respectively, whereas theproblems with ε > 0 are elliptic. In the parabolic and hyperboliccases, thepropagation of singularities appear as an integral property after integratingacross the layers.

Keywords

Singular perturbationsunbounded energypropagation of singularities.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 941 - 963
Copyright
© EDP Sciences, SMAI, 2002

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References

W. Eckhaus, Asymptotic analysis of singular perturbations. North-Holland, Amsterdam (1979).
I.M. Guelfand and G.E. Chilov, Les distributions. Dunod, Paris (1962).
Gérard, P. and Sanchez-Palencia, E., Sensitivity phenomena for certain thin elastic shells with edges. Math. Meth. Appl. Sci. 23 (2000) 379-399. 3.0.CO;2-0>CrossRef
A.M. Il'in, Matching of asymptotic expansions of solutions of boundary value problems. Amer. Math. Soc. (1991).
P. Karamian and J. Sanchez-Hubert, Boundary layers in thin elastic shells with developable middle surface. Eur. J. Mech., A/Solids 21 (2002) 13-47. CrossRef
P. Karamian, J. Sanchez-Hubert and E. Sanchez-Palencia, Propagation of singularities and structure of the layers in shells. Hyperbolic case. Comp. and Structures (to appear).
J.-L. Lions, Perturbations singulières dans les problèmes aux limites et en contrôle optimal. Springer, Berlin (1973).
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1. Dunod, Paris (1968).
Lions, J.-L. and Sanchez-Palencia, E., Sensitivity of certain constrained systems and application to shell theory. J. Math. Pures Appl. 79 (2000) 821-838. CrossRef
Sanchez-Palencia, E., On a singular perturbation going out of the energy space. J. Math. Pures. Appl. 79 (2000) 591-602. CrossRef
E. Sanchez-Palencia, Singular perturbations going out of the energy space. Layers in elliptic and parabolic cases, in Proc. of the 4th european Conference on Elliptic and Parabolic Problems. Rolduc-Gaeta, edited by Bemelmans et al. World Scientific Press (2002).
M.I. Vishik and L. Lusternik, Regular degenerescence and boundary layer for linear differential equations with small parameter.Usp. Mat. Nauk 12 (1957) 1-122.