Appl Math Optim 22:211 226 (1990) Applied Mathematics and Optimization © 1990 Springer-VerlagNew York Inc. Duality Applied to Contact Problems with Friction* A. Klarbring, 1 A. Mikeli6, 2 and M. Shillor 3 IDepartment of Mechanical Engineering, Link6ping Institute of Technology Link6ping, Sweden 2Department of Theoretical Physics, Intstitute "Rudjer Bogkovi6," Zagreb, Yugoslavia 3Department of Mathematical Sciences, Oakland University, Rochester, MI 48309, USA Abstract. The duality theory of Mosco, Capuzzo-Dolcetta, and Matzeu for variational and quasi-variational inequalities is extended. Then it is applied to two problems of contact with friction of an elastic body with a rigid foundation. The more realistic normal compliance condition is used in place of Signorini's condition on the contact surface. 1. Introduction We consider the duality theory for variational and quasi-variational inequalities and apply it to two problems of frictional contact of linearly elastic bodies with surface normal compliance. The dual formulation of the variational inequality find u~V such that Vw~V (Au, w - u) + q~(w) - q~(u) >_ 0 has been studied, essentially, in two ways. The first is the direct application of the Fenchel-Rockafellar duality to the convex function w --* (Au, w) + q)(w). It has been developed, and can be found, in *We (A.M. and M.S.) are grateful for the financial support and hospitality of the Department of Mechanical Engineering in the Linkfping Institute for Technology during our two weeks stay in August 1988. In addition M.S. was partially supported by the Oakland University Faculty Research Award.