On equilibrium of a two-layer elastic structure with a crack in non-coercive case

The paper concerns an analysis of a free boundary problem describing an equilibrium state of a two-layer elastic structure with a crack in a non-coercive case. The non-coercivity of the problem is a consequence of the Neumann boundary conditions on external boundaries. Nonlinear constraints are imposed at the crack faces providing a mutual non-penetration between them. Necessary and sufficient conditions imposed on external forces are found for a solvability of the problem. Limiting passages are investigated as a rigidity of the elastic layer converges to infinity and to zero. Both variational and differential problems formulations are analyzed.