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

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- 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 (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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- 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
- 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. Zbl1006.74064
- 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
- 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). Zbl0165.10801
- 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
- E. Sanchez-Palencia, On a singular perturbation going out of the energy space. J. Math. Pures. Appl.79 (2000) 591-602. Zbl0958.35008
- 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
- 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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