Optimal shapes of cracks in a viscoelastic body

We consider an optimal control problem for equations describing the quasistatic deformation of a linear viscoelastic body. There is a crack in the body, and displacements of opposite faces of the crack are constrained by the nonpenetration condition. The continuous dependence of the solution to the equilibrium problem on the shape of the crack is established. In particular, we prove the existence of a shape for which the crack opening is minimal