zbMATH — the first resource for mathematics

Various kinds of sensitive singular perturbations. (English) Zbl 1153.35011
This paper concerns singularly perturbed abstract linear equations in variational form. The results are applied to boundary value problems for PDEs having highly pathological asymptotic behavior such that the limit problems are out of classical PDE theory (because the limit bilinear forms are continuous and coercive on very large spaces, not contained in distribution spaces, only). Two examples from this shell theory are presented. The highly pathological asymptotic behavior of the first (resp. second) example comes from a boundary condition not satisfying the Shapiro-Lopatinski condition (resp. from an edge in the shell).
The paper is mainly a review of the authors’ previous results. It includes interesting remarks and comments, in particular on spaces of analytical functionals (out of distribution spaces) and on complexification.

35B25 Singular perturbations in context of PDEs
35G15 Boundary value problems for linear higher-order PDEs
46F15 Hyperfunctions, analytic functionals
74K25 Shells
Full Text: DOI Numdam EuDML
[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., 17, 35-92, (1964) · Zbl 0123.28706
[2] Babuska, I.; Suri, M., On locking and robustness in the finite element method, SIAM J. Num. Anal., 29, 1261-1293, (1992) · 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., 323, 7, 835-840, (1996) · Zbl 0864.47004
[4] Courant, R.; Hilbert, D., Methods of mathematical physics. Vol. II, (1989), John Wiley & Sons Inc., New York · 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., 9, 6, 996-1019, (1978) · Zbl 0406.46033
[6] Egorov, Y. V.; Schulze, B.-W., Pseudo-differential operators, singularities, applications, 93, (1997), Birkhäuser Verlag, Basel · Zbl 0877.35141
[7] Erdélyi, A., Asymptotic expansions, (1956), Dover Publications Inc., New York · Zbl 0070.29002
[8] Gerard, P.; Sanchez-Palencia, E., Sensitivity phenomena for certain thin elastic shells with edges, Math. Methods Appl. Sci., 23, 4, 379-399, (2000) · Zbl 0989.74047
[9] Gindikin, S. G.; Volevich, L. R., Partial differential equations, 3 (Russian), The Cauchy problem, 5-98, 220, (1988), Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow · Zbl 0738.35002
[10] Guelfand, I. M.; Chilov, G. E., Les distributions, (1962), Dunod, Paris · Zbl 0115.10102
[11] Huet, D., Phénomènes de perturbation singulière dans LES problèmes aux limites, Ann. Inst. Fourier. Grenoble, 10, 61-150, (1960) · Zbl 0128.32904
[12] Komech, A. I., Partial differential equations, II, 31, Linear partial differential equations with constant coefficients, 121-255, (1994), Springer, Berlin · Zbl 0805.35001
[13] Lions, J.-L., Perturbations singulières dans les problèmes aux limites et en contrôle optimal, (1973), Springer-Verlag, Berlin · Zbl 0268.49001
[14] Lions, J.-L.; Sanchez-Palencia, E., Partial differential equations and functional analysis, 22, Problèmes sensitifs et coques élastiques minces, 207-220, (1996), Birkhäuser Boston, Boston, MA · Zbl 0857.35033
[15] Meunier, N.; Sanchez-Palencia, E., Sensitive versus classical perturbation problem via Fourier transform, Math. Models Methods Appl. Sci., (2007) · Zbl 1246.35026
[16] Sanchez-Hubert, J.; Sanchez-Palencia, E., Vibration and coupling of continuous systems, (1989), Springer-Verlag, Berlin · Zbl 0698.70003
[17] Sanchez-Hubert, J.; Sanchez-Palencia, E., Coques élastiques minces, (1997), Masson, Paris · Zbl 0881.73001
[18] Sanchez-Hubert, J.; Sanchez-Palencia, E., Partiall Diff. Eq. in Micostructures, Singular perturbations with non-smooth limit and finite element approximation of layers for model problems of shells, 207-226, (2001), F. Ali Mehmet, J. Von Bellow and S. Nicaise ed., Marcel Dekker · Zbl 1079.35010
[19] Sanchez-Palencia, E., Asymptotic and spectral properties of a class of singular-stiff problems, J. Math. Pures Appl. (9), 71, 5, 379-406, (1992) · Zbl 0833.47011
[20] Sanchez-Palencia, E.; De Souza, C., Fluid Mechanics, Complexification phenomena in certain singular perturbations, 363-379, (2004), F. J. Higuera, J. Jimenez, J. M. Vegan, ed., CIMNE, Barcelona
[21] Sanchez-Palencia, E.; De Souza, C., Complexification phenomenon in an example of sensitive singular perturbation, C. R. Acad. Sci. Paris Sér. II. Méc., 332, 8, 605-612, (2004) · Zbl 1239.35012
[22] Sanchez-Palencia, E.; De Souza, C., Complexification in singular perturbations and their approximation, Int. J. Multiscale Comput. Eng., 3, 481-498, (2006)
[23] Schwartz, L., Théorie des distributions. Tome I, (1957), Hermann & Cie., Paris · Zbl 0078.11003
[24] Smirnov, V. I., A course of higher mathematics. Vol. III. Part one. Linear algebra, (1964), Pergamon Press, Oxford · Zbl 0121.25904
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.