Various kinds of sensitive singular perturbations
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 ε when the limit process ε0 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 ε0.

@article{AMBP_2007__14_2_199_0,
     author = {Meunier, Nicolas and Sanchez-Hubert, Jacqueline and Sanchez-Palencia, \'Evariste},
     title = {Various kinds of sensitive singular perturbations},
     journal = {Annales Math\'ematiques Blaise Pascal},
     pages = {199--242},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {14},
     number = {2},
     year = {2007},
     doi = {10.5802/ambp.233},
     mrnumber = {2369872},
     zbl = {1153.35011},
     language = {en},
     url = {www.numdam.org/item/AMBP_2007__14_2_199_0/}
}
Meunier, Nicolas; Sanchez-Hubert, Jacqueline; Sanchez-Palencia, Évariste. Various kinds of sensitive singular perturbations. Annales Mathématiques Blaise Pascal, Tome 14 (2007) no. 2, pp. 199-242. doi : 10.5802/ambp.233. http://www.numdam.org/item/AMBP_2007__14_2_199_0/

[1] Agmon, S.; Douglis, A.; Nirenberg, L. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math., Volume 17 (1964), pp. 35-92 | Article | MR 162050 | Zbl 0123.28706

[2] Babuska, I.; Suri, M. On locking and robustness in the finite element method, SIAM J. Num. Anal., Volume 29 (1992), pp. 1261-1293 | Article | MR 1182731 | Zbl 0763.65085

[3] Caillerie, D. Étude générale d’un type de problèmes raides et de perturbation singulière, C. R. Acad. Sci. Paris Sér. I, Math., Volume 323 (1996) no. 7, pp. 835-840 | Zbl 0864.47004

[4] Courant, R.; Hilbert, D. Methods of mathematical physics. Vol. II, Wiley Classics Library, John Wiley & Sons Inc., New York, 1989 (Partial differential equations, Reprint of the 1962 original, A Wiley-Interscience Publication) | MR 1013360 | Zbl 0729.35001

[5] De Roever, J. W. Analytic representations and Fourier transforms of analytic functionals in Z carried by the real space, SIAM J. Math. Anal., Volume 9 (1978) no. 6, pp. 996-1019 | Article | MR 512506 | Zbl 0406.46033

[6] Egorov, Y. V.; Schulze, B.-W. Pseudo-differential operators, singularities, applications, Operator Theory: Advances and Applications, Volume 93, Birkhäuser Verlag, Basel, 1997 | MR 1443430 | Zbl 0877.35141

[7] Erdélyi, A. Asymptotic expansions, Dover Publications Inc., New York, 1956 | MR 78494 | Zbl 0070.29002

[8] Gerard, P.; Sanchez-Palencia, E. Sensitivity phenomena for certain thin elastic shells with edges, Math. Methods Appl. Sci., Volume 23 (2000) no. 4, pp. 379-399 | Article | MR 1740321 | Zbl 0989.74047

[9] Gindikin, S. G.; Volevich, L. R. The Cauchy problem, Partial differential equations, 3 (Russian) (Itogi Nauki i Tekhniki), Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988, p. 5-98, 220 | MR 1133456 | Zbl 0738.35002

[10] Guelfand, I. M.; Chilov, G. E. Les distributions, Collection Universitaire de Mathématiques, VIII, Dunod, Paris, 1962

[11] Huet, D. Phénomènes de perturbation singulière dans les problèmes aux limites, Ann. Inst. Fourier. Grenoble, Volume 10 (1960), pp. 61-150 | Article | Numdam | MR 118968 | Zbl 0128.32904

[12] Komech, A. I. Linear partial differential equations with constant coefficients, Partial differential equations, II (Encyclopaedia Math. Sci.) Volume 31, Springer, Berlin, 1994, pp. 121-255 | MR 1364201 | Zbl 0805.35001

[13] Lions, J.-L. Perturbations singulières dans les problèmes aux limites et en contrôle optimal, Springer-Verlag, Berlin, 1973 (Lecture Notes in Mathematics, Vol. 323) | MR 600331 | Zbl 0268.49001

[14] Lions, J.-L.; Sanchez-Palencia, E. Problèmes sensitifs et coques élastiques minces, Partial differential equations and functional analysis (Progr. Nonlinear Differential Equations Appl.) Volume 22, Birkhäuser Boston, Boston, MA, 1996, pp. 207-220 | MR 1399133 | Zbl 0857.35033

[15] Meunier, N.; Sanchez-Palencia, E. Sensitive versus classical perturbation problem via Fourier transform, Math. Models Methods Appl. Sci. (2007 (to appear)) | MR 2271599 | Zbl 05123883

[16] Sanchez-Hubert, J.; Sanchez-Palencia, E. Vibration and coupling of continuous systems, Springer-Verlag, Berlin, 1989 (Asymptotic methods) | MR 996423 | Zbl 0698.70003

[17] Sanchez-Hubert, J.; Sanchez-Palencia, E. Coques élastiques minces, Masson, Paris, 1997 (Propriétés asymptotiques) | Zbl 0881.73001

[18] Sanchez-Hubert, J.; Sanchez-Palencia, E. Singular perturbations with non-smooth limit and finite element approximation of layers for model problems of shells, Partiall Diff. Eq. in Micostructures, F. Ali Mehmet, J. Von Bellow and S. Nicaise ed., Marcel Dekker, 2001, pp. 207-226 | MR 1824574 | Zbl 1079.35010

[19] Sanchez-Palencia, E. Asymptotic and spectral properties of a class of singular-stiff problems, J. Math. Pures Appl. (9), Volume 71 (1992) no. 5, pp. 379-406 | MR 1191581 | Zbl 0833.47011

[20] Sanchez-Palencia, E.; De Souza, C. Complexification phenomena in certain singular perturbations, Fluid Mechanics, F. J. Higuera, J. Jimenez, J. M. Vegan, ed., CIMNE, Barcelona, 2004, pp. 363-379

[21] Sanchez-Palencia, E.; De Souza, C. Complexification phenomenon in an example of sensitive singular perturbation, C. R. Acad. Sci. Paris Sér. II. Méc., Volume 332 (2004) no. 8, pp. 605-612

[22] Sanchez-Palencia, E.; De Souza, C. Complexification in singular perturbations and their approximation, Int. J. Multiscale Comput. Eng., Volume 3 (2006), pp. 481-498

[23] Schwartz, L. Théorie des distributions. Tome I, Actualités Sci. Ind., no. 1245. Publ. Inst. Math. Univ. Strasbourg, Hermann & Cie., Paris, 1957 | MR 35918 | Zbl 0078.11003

[24] Smirnov, V. I. A course of higher mathematics. Vol. III. Part one. Linear algebra, Translated by D. E. Brown. Translation edited by I. N. Sneddon, Pergamon Press, Oxford, 1964 | Zbl 0121.25904