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 (2010)

  • 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 (2010): 941-963. <http://eudml.org/doc/90680>.

@article{Sanchez2010,
abstract = { 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. },
author = {Sanchez–Palencia, E.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Singular perturbations; unbounded energy; propagation of singularities.; propagation of singularities; distribution right hand sides},
language = {eng},
month = {3},
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/90680},
volume = {8},
year = {2010},
}

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
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 941
EP - 963
AB - 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.
LA - eng
KW - Singular perturbations; unbounded energy; propagation of singularities.; propagation of singularities; distribution right hand sides
UR - http://eudml.org/doc/90680
ER -

References

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  1. W. Eckhaus, Asymptotic analysis of singular perturbations. North-Holland, Amsterdam (1979).  Zbl0421.34057
  2. I.M. Guelfand and G.E. Chilov, Les distributions. Dunod, Paris (1962).  Zbl0115.10102
  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. A.M. Il'in, Matching of asymptotic expansions of solutions of boundary value problems. Amer. Math. Soc. (1991).  
  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. 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).  Zbl1023.74030
  7. J.-L. Lions, Perturbations singulières dans les problèmes aux limites et en contrôle optimal. Springer, Berlin (1973).  
  8. J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1. Dunod, Paris (1968).  Zbl0165.10801
  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. E. Sanchez-Palencia, On a singular perturbation going out of the energy space. J. Math. Pures. Appl.79 (2000) 591-602.  Zbl0958.35008
  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. M.I. Vishik and L. Lusternik, Regular degenerescence and boundary layer for linear differential equations with small parameter.Usp. Mat. Nauk12 (1957) 1-122.  

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