Tykhonov well-posedness of a frictionless unilateral contact problem

Abstract

We consider a frictionless contact problem, Problem $\mathcal{P}$ , for elastic materials. The process is assumed to be static and the contact is modelled with unilateral constraints. We list the assumptions on the data and derive a variational formulation of the problem, Problem ${\mathcal{P}}_V$ . Then we consider a perturbation of Problem $\mathcal{P}$ , which could be frictional, governed by a small parameter $\epsilon$ . This perturbation leads in a natural way to a family of sets $\{\mathrm{\Omega }(\epsilon ){\}}_{\epsilon >0}$ . We prove that Problem ${\mathcal{P}}_V$ is well-posed in the sense of Tykhonov with respect to the family $\{\mathrm{\Omega }(\epsilon ){\}}_{\epsilon >0}$ . The proof is based on arguments of monotonicity, pseudomonotonicity and various estimates. We extend these results to a time-dependent version of Problem $\mathcal{P}$ . Finally, we provide examples and mechanical interpretation of our well-posedness results, which, in particular, allow us to establish the link between the weak solutions of different contact models.