Qualitative properties of solutions of the unilateral elliptic problem and of the Signorini problem for the Lame system of equations are considered. A preliminary analysis reduces each of the nonlinear problems to a mixed linear problem and to additional conditions (in the form of inequalities) on the part of the boundary. In each case, it is shown that if a perturbed unknown boundary involves the true one, i.e. Γ ̃ c ⊂Γ c , then the penetration condition is failed on Γ c ⧹ Γ ̃ c . Conversely, if Γ c ⊂ Γ ̃ c , then the second additional condition in the form of inequality is failed on Γ ̃ c ⧹Γ c . The obtained results for the Signorini type problems have a precise physical interpretation in the case of contact problems.