Properties of singnorini's problem solution with perturbation of unknown boundary and application to contact problems

Abstract

Qualitative properties of an elliptic unilateral problems for a scalar elliptic equationsand for the Lame system of equations are considered. A preliminary analysis reduces each of the nonlinear problems to a mixed linear problem and to an additional (in the form of inequality) conditions on the part of boundary. In each case, it is shown that if a perturbed unknown boundary involves the true one, i.e. , then a penetration condition is failed on Conversely, if, then the second additional condition in the form of inequality is failed on . The obtained results for unilateral elliptic problems have a precise physical interpretation in the case of contact problems. Using obtained results a new approach to approximate unilateral elliptic problems is presented: a sblution of corresponding unilateral problem is sought as a sequence of linear mixed problem solutions. A demonstration of obtained results is illustrated also numerically on solved examples.