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On ill-posedness of free-boundary problems for highly compressible two-dimensional elastic bodies

Authors: Yu. V. Egorov and E. Sanchez-Palencia
Original publication: Algebra i Analiz, tom 22 (2010), nomer 6.
Journal: St. Petersburg Math. J. 22 (2011), 913-926
MSC (2010): Primary 35R25; Secondary 74B99
Published electronically: August 18, 2011
MathSciNet review: 2760087
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Abstract: 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.

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Additional Information

Yu. V. Egorov
Affiliation: Laboratoire MIP, Université Paul Sabatier, 118 route de Narbonne, Toulouse 31062, France

E. Sanchez-Palencia
Affiliation: Laboratoire de Modélisation en Méchanique 4, Université Pierre et Marie Curie, place Jussieu, case 162, Paris 75252, France

Keywords: Two-dimensional elasticity, negative Poisson ratio, elliptic boundary value problems
Received by editor(s): June 29, 2010
Published electronically: August 18, 2011
Dedicated: To V.M.Babich on the occasion of his 80th birthday
Article copyright: © Copyright 2011 American Mathematical Society