The limit behavior of the solutions of Signorini's type-likeproblems in periodically perforated domains with periodε is studied. The main feature of this limit behaviour isthe existence of a critical size of the perforations thatseparates different emerging phenomena as ε → 0. In the critical case, it is shown that Signorini's problemconverges to a problem associated to a new operator whichis the sum of a standard homogenized operator and an extra zeroorder term (“strange term”) coming from the geometry; itsappearance is due to the special size of the holes. The limitproblem captures the two sources of oscillations involved in thiskind of free boundary-value problems, namely, those arising fromthe size of the holes and those due to the periodic inhomogeneityof the medium. The main ingredient of the method used in the proofis an explicit construction of suitable test functions whichprovide a good understanding of the interactions between the abovementioned sources of oscillations.

We consider the identification of a distributed parameter in an ellipticvariational inequality. On the basis of an optimal control problemformulation, the application of a primal-dual penalizationtechnique enables us to prove the existenceof multipliers giving a first order characterization of the optimal solution.Concerning the parameter we consider differentregularity requirements. For the numerical realization we utilize a complementarity function,which allows us to rewrite the optimality conditions as a set of equalities.Finally, numerical results obtained from a least squares type algorithmemphasize the feasibility of our approach.