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Communicated by B. BrosowskiUsing Ball's approach to non-linear elasticity, and in particular his concept of polyconvexity, we treat a unilateral three-dimensional contact problem for a hyperelastic body under volume and surface forcs. Here the unilateral constraint is described by a sublinear function which can model the contact with a rigid convex cone. We obtain a solution to this generally non-convex, semicoercive Signorini problem as a limit of solutions of related energy minimization problems involving friction normal to the contact surface where the friction coefficient goes to infinity. Thus we extend an approximation result of Duvaut and Lions for lineadastic unilateral contact problems to finite deformations and to a class of non-linear elastic materials including the material models of Ogden and of Mooney-Rivlin for rubberlike materials.Moreover, the underlying penalty method is shown to be exact, that is a sufficiently large friction coefficient in the auxiliary energy minimization problems suffioes to produce a solution of the original unilateral problem, provided a Lagrange multiplier to the unilateral constraint exists.

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Abstract. In this contribution we introduce to the topic of Random Variational Inequalities (RVI) and present some of our recent results in this field. We show how the theory of monotone RVI, where random variables occur both in the operator and the constraints set, can be applied to model nonlinear equilibrium problems under uncertainty arising from economics and operations research, including migration and transportation science. In particular we treat Wardrop equilibria in traffic networks. We describe an approximation procedure for the statistical quantities connected to the equilibrium solution and illustrate this procedure by means of some small sized numerical examples.