On the solvability of a variational problem about phase transitions in continuum mechanics

We study a variational problem about phase transitions in continuum mechanics under the condition that the surface tension coefficient vanishes. A homogeneous isotropic two-phase elastic medium occupies a ball-shaped domain, the zero displacement field is fixed on the boundary of this domain, and a spherically symmetric force field acts on the medium. The solvability of this problem is established. As is shown, if a force field is nonzero almost everywhere, then the problem has only spherically symmetric solutions. Bibliography: 9 titles. Translated from Problemy Matematicheskogo Analiza, No. 38, December 2008, pp. 61–71.