Geodesic
On ill-posedness of free-boundary problems for highly compressible two-dimensional elastic bodies
Algebra i analiz, Tome 22 (2010) no. 6, pp. 91-108.

Voir la notice de l'article provenant de la source Math-Net.Ru

Some problems of elasticity theory related to highly compressible two-dimensional elastic bodies are considered. Such problems arise in real elasticity and pertain to some materials having negative Poisson ratio. The common feature of such problems is the presence of a small parameter $\varepsilon$. If $\varepsilon>0$, the corresponding equations are elliptic and the boundary data obey the Shapiro–Lopatinsky condition. If $\varepsilon=0$, this condition is violated and the problem may fail to be solvable in distribution spaces. The rather difficult passing to the limit is studied.
Keywords: two-dimensional elasticity, negative Poisson ratio, elliptic boundary value problems. 
@article{AA_2010_22_6_a4,
     author = {Yu. V. Egorov and E. Sanchez-Palencia},
     title = {On ill-posedness of free-boundary problems for highly compressible two-dimensional elastic bodies},
     journal = {Algebra i analiz},
     pages = {91--108},
     publisher = {mathdoc},
     volume = {22},
     number = {6},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AA_2010_22_6_a4/}
}
TY  - JOUR
AU  - Yu. V. Egorov
AU  - E. Sanchez-Palencia
TI  - On ill-posedness of free-boundary problems for highly compressible two-dimensional elastic bodies
JO  - Algebra i analiz
PY  - 2010
SP  - 91
EP  - 108
VL  - 22
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AA_2010_22_6_a4/
LA  - en
ID  - AA_2010_22_6_a4
ER  - 
%0 Journal Article
%A Yu. V. Egorov
%A E. Sanchez-Palencia
%T On ill-posedness of free-boundary problems for highly compressible two-dimensional elastic bodies
%J Algebra i analiz
%D 2010
%P 91-108
%V 22
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AA_2010_22_6_a4/
%G en
%F AA_2010_22_6_a4
Yu. V. Egorov; E. Sanchez-Palencia. On ill-posedness of free-boundary problems for highly compressible two-dimensional elastic bodies. Algebra i analiz, Tome 22 (2010) no. 6, pp. 91-108. http://geodesic.mathdoc.fr/item/AA_2010_22_6_a4/

[1] Egorov Yu. V., Shubin M. A., “Lineinye differentsialnye uravneniya s chastnymi proizvodnymi. Osnovy klassicheskoi teorii”, Differentsialnye uravneniya s chastnymi proizvodnymi – 1, Itogi nauki i tekhn. Sovrem. probl. mat. Fundam. napravleniya, 30, VINITI, M., 1988, 5–255 | MR | Zbl

[2] Landau L. D., Lifshits E. M., Teoreticheskaya fizika, v. 7, Teoriya uprugosti, 4-e izd., Nauka, M., 1987 | MR | Zbl

[3] Béchet F., Millet O., Sanchez-Palencia E., “Singular perturbations generating complexification phenomena for elliptic shells”, Comput. Mech., 43 (2009), 207–221 | DOI | MR | Zbl

[4] Egorov Yu. V., Meunier N., Sanchez-Palencia E., “Rigorous and heuristic treatment of certain sensitive singular perturbations”, J. Math. Pures Appl. (9), 88 (2007), 123–147 | MR | Zbl

[5] Egorov Yu. V., Meunier N., Sanchez-Palencia E., “Rigorous and heuristic treatment of sensitive singular perturbations arising in elliptic shells, II”, Around the Research of V. Mazýa, Springer-Verlag, Berlin, 2010, 159–202 | Zbl

[6] Almgren R. F., “An anisotropic three-dimensional structure with Poisson's ration $=-1$”, J. Elasticity, 15 (1985), 427–430 | DOI

[7] Lakes R., “Foam structures with negative Poisson ratio”, Science AAAS, 235 (1987), 1038–1040

[8] Babich V. M., Buldyrev V. S., Asimptoticheskie metody v zadachakh difraktsii korotkikh voln. Metod etalonnykh zadach, Nauka, M., 1972 | MR