# American Institute of Mathematical Sciences

December  2011, 31(4): 1293-1305. doi: 10.3934/dcds.2011.31.1293

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

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

Received  June 2010 Revised  October 2010 Published  September 2011

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 $|\xi|\asymp \log (\varepsilon^{-1})$.
Citation: Youri V. Egorov, Evariste Sanchez-Palencia. Remarks on certain singular perturbations with ill-posed limit in shell theory and elasticity. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1293-1305. doi: 10.3934/dcds.2011.31.1293
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