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Some problems of elasticity theory related to highly compressible twodimensional 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 ε. If ε > 0, the corresponding equations are elliptic and the boundary data obey the Shapiro–Lopatinsky condition. If ε = 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. Introduction Elliptic boundary value problems are nowadays well studied and take a very important place in the modern theory of partial differential equations. Their theory is based on the so-called Shapiro–Lopatinsky conditions (SL) that give an algebraic reformulation of the normal solvability of boundary value problems. (See, for example, [1, Chapter 2, 2].) Usually, violation of SL leads to a very difficult problem and demands new methods. Here we consider some examples of this kind, arising in real elasticity problems (see also [3]–[5] and the references therein). These examples pertain to some materials having negative Poisson ratio (see [2, 6, 7]). Actually, they are merely models, but they show the specific character of the problems in question and the nature of the results in real situations. Moreover, such situations arise in many other problems of mathematical physics. The principal feature of these problems is the necessity to consider elliptic boundary value problems with the Shapiro–Lopatinsky condition violated. This implies the unsolvability of the problems in distribution spaces. To solve such problems, it is necessary to consider generalized functions with exponentially growing Fourier transforms. The common property of such problems is the fact that they involve a small parameter ε and reduce to an equation of the form (A + εA)w = f with A generically smoothing and A elliptic. The main role in the asymptotic expansion of solutions as ε tends to 0 is played by the Fourier components with frequencies that are O(log(ε−1)), and the convergence to the limit is rather slow. Similar phenomena were observed by Babich and Buldyrev in the book [8]. 2010 Mathematics Subject Classification. Primary 35R25, Secondary 74B99.

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