Qualitative behaviour of solutions of unilateral elliptic problems with perturbing the unknown boundary I. The theory

摘要

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. Γ̃c⊂Γc, then the penetration condition is failed on Γc⧹Γ̃c. Conversely, if Γc⊂Γ̃c, then the second additional condition in the form of inequality is failed on Γ̃c⧹Γc. The obtained results for the Signorini type problems have a precise physical interpretation in the case of contact problems.