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

Abstract

Some problems of elasticity and shell theory are considered. Thecommon feature of these problems is the presence of a smallparameter $\varepsilon$. If $\varepsilon>0$ the correspondingequations are elliptic and the boundary conditions satisfy theShapiro - Lopatinsky condition. When $\varepsilon=0$, this conditionis violated and the problem can be non-solvable in the distributionspaces. The rather difficult passing to the limit is studied usingthe related Cauchy problem for elliptic equations. This approachallows to show that the most important is the transition zone wherethe frequencies $|\xi|\asymp \log (\varepsilon^{-1})$.