Effect of the incompressibility condition on the stability of the solutions of ill-posed problems of the theory of elasticity

One considers the energy functional with density depending only on the first invariant of the strain tensor. Under natural constraints on the density and in the absence of exterior actions, the zero displacements yield a minimum to the energy functional both for the compressible and the incompressible media. In the paper one proves the isolation of this minimum for incompressible media and its nonisolation for compressible media.Translated from Problemy Matemticheskogo Analiza, No. 10, pp. 62–71, 1986.