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

Laboratoire de Modélisation en Mécanique, CNRS-Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05, France; sanchez@lmm.jussieu.fr.

Received: 7 January 2002

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.