We consider singular perturbation variational problems depending on a small parameter . The right hand side is such that the energy does not remain bounded as . 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 are elliptic. In the parabolic and hyperbolic cases, the propagation of singularities appear as an integral property after integrating across the layers.
Classification : 35A35, 35B25, 35B40
Mots clés : singular perturbations, unbounded energy, propagation of singularities
@article{COCV_2002__8__941_0,
author = {Sanchez-Palencia, E.},
title = {On the structure of layers for singularly perturbed equations in the case of unbounded energy},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {941--963},
publisher = {EDP-Sciences},
volume = {8},
year = {2002},
doi = {10.1051/cocv:2002043},
zbl = {1070.35005},
language = {en},
url = {www.numdam.org/item/COCV_2002__8__941_0/}
}
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, Tome 8 (2002) , pp. 941-963. doi : 10.1051/cocv:2002043. http://www.numdam.org/item/COCV_2002__8__941_0/
[1] , Asymptotic analysis of singular perturbations. North-Holland, Amsterdam (1979). | MR 553107 | Zbl 0421.34057
[2] and, Les distributions. Dunod, Paris (1962). | MR 132390 | Zbl 0115.10102
[3] and, Sensitivity phenomena for certain thin elastic shells with edges. Math. Meth. Appl. Sci. 23 (2000) 379-399. | Zbl 0989.74047
[4] , Matching of asymptotic expansions of solutions of boundary value problems. Amer. Math. Soc. (1991).
[5] and, Boundary layers in thin elastic shells with developable middle surface. Eur. J. Mech., A/Solids 21 (2002) 13-47. | Zbl 1006.74064
[6] , and, Propagation of singularities and structure of the layers in shells. Hyperbolic case. Comp. and Structures (to appear).
[7] , Perturbations singulières dans les problèmes aux limites et en contrôle optimal. Springer, Berlin (1973). | MR 600331 | Zbl 0268.49001
[8] and, Problèmes aux limites non homogènes et applications, Vol. 1. Dunod, Paris (1968). | MR 247243 | Zbl 0165.10801
[9] and, Sensitivity of certain constrained systems and application to shell theory. J. Math. Pures Appl. 79 (2000) 821-838. | Zbl 1013.74050
[10] , On a singular perturbation going out of the energy space. J. Math. Pures. Appl. 79 (2000) 591-602. | Zbl 0958.35008
[11] , 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). | Zbl 1033.35010
[12] and, Regular degenerescence and boundary layer for linear differential equations with small parameter.Usp. Mat. Nauk 12 (1957) 1-122. | Zbl 0087.29602
