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

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.