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

Egorov, Yu.; Sanchez-Palencia, E.

doi:10.1090/S1061-0022-2011-01176-8

On ill-posedness of free-boundary problems for highly compressible two-dimensional elastic bodies
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by Yu. V. Egorov and E. Sanchez-Palencia

St. Petersburg Math. J. 22 (2011), 913-926

DOI: https://doi.org/10.1090/S1061-0022-2011-01176-8

Published electronically: August 18, 2011

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Abstract:

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.

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