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

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

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

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topSanchez-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 >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 >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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- [2] I.M. Guelfand and G.E. Chilov, Les distributions. Dunod, Paris (1962). Zbl0115.10102MR132390
- [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).
- [7] J.-L. Lions, Perturbations singulières dans les problèmes aux limites et en contrôle optimal. Springer, Berlin (1973). Zbl0268.49001MR600331
- [8] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1. Dunod, Paris (1968). Zbl0165.10801MR247243
- [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. Nauk 12 (1957) 1-122. Zbl0087.29602

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