Weak solvability via Lagrange multipliers for contact problems involving multi-contact zones

Abstract

We investigate the behaviour of an elastic body which is in frictional contact with a foundation on a part of the boundary, and which can come into contact with a rigid obstacle on another part of the boundary. We associate this physical setting with two mechanical models. Every model is mathematically described by a boundary value problem which consists of a system of partial differential equations associated with a displacement condition, a traction condition, a frictional contact condition and a frictionless unilateral contact condition. In both models the unilateral contact is described by Signorini’s condition with non-zero gap. The difference between the models is given by the frictional condition we use. In the first model we use a condition with prescribed normal stress. In the second one, we use a frictional bilateral contact condition. The weak solvability of the boundary value problems we propose herein relies on an abstract generalized saddle point problem. Abstract existence, uniqueness and boundedness results as well as abstract convergence results of a regularization are established. Then, we discuss the existence, the uniqueness, the boundedness and the approximation of the weak solutions based on the abstract results.