Remarks on certain singular perturbations with ill-posed limit in shell theory and elasticity

Youri V. Egorov; Evariste Sanchez-Palencia

doi:10.3934/dcds.2011.31.1293

Article Contents

2011, Volume 31, Issue 4: 1293-1305. Doi: 10.3934/dcds.2011.31.1293

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Remarks on certain singular perturbations with ill-posed limit in shell theory and elasticity

Youri V. Egorov^1, and
Evariste Sanchez-Palencia^2,

1.
IMT, Université Paul Sabatier, 118, route de Narbonne, Toulouse, 31062
2.
Laboratoire de Modélisation en Mécanique, Université Pierre et Marie Curie 4, place Jussieu, Paris, 75252

Received: May 31, 2010

Revised: September 30, 2010

Published: September 2011

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Abstract

Some problems of elasticity and shell theory are considered. The common feature of these problems is the presence of a small parameter $\varepsilon$. If $\varepsilon>0$ the corresponding equations are elliptic and the boundary conditions satisfy the Shapiro - Lopatinsky condition. When $\varepsilon=0$, this condition is violated and the problem can be non-solvable in the distribution spaces. The rather difficult passing to the limit is studied using the related Cauchy problem for elliptic equations. This approach allows to show that the most important is the transition zone where the frequencies $|\xi|\asymp \log (\varepsilon^{-1})$.

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Mathematics Subject Classification: Primary: 35J25; Secondary: 74B05.

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References

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