Various kinds of sensitive singular perturbations

Meunier, Nicolas; Sanchez-Hubert, Jacqueline; Sanchez-Palencia, Évariste

Annales mathématiques Blaise Pascal, Tome 14 (2007) no. 2, pp. 199-242.

We consider variational problems of P. D. E. depending on a small parameter $\epsilon$ when the limit process $\epsilon \downarrow 0$ implies vanishing of the higher order terms. The perturbation problem is said to be sensitive when the energy space of the limit problem is out of the distribution space, so that the limit problem is out of classical theory of P. D. E. We present here a review of the subject, including abstract convergence theorems and two very different model problems (the second one is presented for the first time). For each one we prove the sensitive character and we give a formal asymptotics for the behavior $\epsilon \downarrow 0$.

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