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

E. Sanchez-Palencia

ESAIM: Control, Optimisation and Calculus of Variations (2002)

  • Volume: 8, page 941-963
  • ISSN: 1292-8119

Abstract

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We consider singular perturbation variational problems depending on a small parameter ε . The right hand side is such that the energy does not remain bounded as ε → 0 . The asymptotic behavior involves internal layers where most of the energy concentrates. Three examples are addressed, with limits elliptic, parabolic and hyperbolic respectively, whereas the problems with ε > 0 are elliptic. In the parabolic and hyperbolic cases, the propagation of singularities appear as an integral property after integrating across the layers.

How to cite

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Sanchez-Palencia, E.. "On the structure of layers for singularly perturbed equations in the case of unbounded energy." ESAIM: Control, Optimisation and Calculus of Variations 8 (2002): 941-963. <http://eudml.org/doc/245292>.

@article{Sanchez2002,
abstract = {We consider singular perturbation variational problems depending on a small parameter $ \varepsilon $. The right hand side is such that the energy does not remain bounded as $ \varepsilon \rightarrow 0 $. The asymptotic behavior involves internal layers where most of the energy concentrates. Three examples are addressed, with limits elliptic, parabolic and hyperbolic respectively, whereas the problems with $ \varepsilon &gt;0 $ are elliptic. In the parabolic and hyperbolic cases, the propagation of singularities appear as an integral property after integrating across the layers.},
author = {Sanchez-Palencia, E.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {singular perturbations; unbounded energy; propagation of singularities; distribution right hand sides},
language = {eng},
pages = {941-963},
publisher = {EDP-Sciences},
title = {On the structure of layers for singularly perturbed equations in the case of unbounded energy},
url = {http://eudml.org/doc/245292},
volume = {8},
year = {2002},
}

TY - JOUR
AU - Sanchez-Palencia, E.
TI - On the structure of layers for singularly perturbed equations in the case of unbounded energy
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2002
PB - EDP-Sciences
VL - 8
SP - 941
EP - 963
AB - We consider singular perturbation variational problems depending on a small parameter $ \varepsilon $. The right hand side is such that the energy does not remain bounded as $ \varepsilon \rightarrow 0 $. The asymptotic behavior involves internal layers where most of the energy concentrates. Three examples are addressed, with limits elliptic, parabolic and hyperbolic respectively, whereas the problems with $ \varepsilon &gt;0 $ are elliptic. In the parabolic and hyperbolic cases, the propagation of singularities appear as an integral property after integrating across the layers.
LA - eng
KW - singular perturbations; unbounded energy; propagation of singularities; distribution right hand sides
UR - http://eudml.org/doc/245292
ER -

References

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  1. [1] W. Eckhaus, Asymptotic analysis of singular perturbations. North-Holland, Amsterdam (1979). Zbl0421.34057MR553107
  2. [2] I.M. Guelfand and G.E. Chilov, Les distributions. Dunod, Paris (1962). Zbl0115.10102MR132390
  3. [3] P. Gérard and E. Sanchez–Palencia, Sensitivity phenomena for certain thin elastic shells with edges. Math. Meth. Appl. Sci. 23 (2000) 379-399. Zbl0989.74047
  4. [4] A.M. Il’in, Matching of asymptotic expansions of solutions of boundary value problems. Amer. Math. Soc. (1991). 
  5. [5] 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. Zbl1006.74064
  6. [6] 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). 
  7. [7] J.-L. Lions, Perturbations singulières dans les problèmes aux limites et en contrôle optimal. Springer, Berlin (1973). Zbl0268.49001MR600331
  8. [8] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1. Dunod, Paris (1968). Zbl0165.10801MR247243
  9. [9] J.-L. Lions and E. Sanchez–Palencia, Sensitivity of certain constrained systems and application to shell theory. J. Math. Pures Appl. 79 (2000) 821-838. Zbl1013.74050
  10. [10] E. Sanchez–Palencia, On a singular perturbation going out of the energy space. J. Math. Pures. Appl. 79 (2000) 591-602. Zbl0958.35008
  11. [11] 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). Zbl1033.35010
  12. [12] 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. Zbl0087.29602

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