A class of variational-hemivariational inequalities for Bingham type fluids

In this paper we investigate a new class of elliptic variational–hemivariational inequal ities without the relaxed monotonicity condition of the generalized subgradient. The inequality describes the mathematical model of the steady state flow of incompress ible fluid of Bingham type in a bounded domain. The boundary condition represents a generalization of the no leak condition, and a multivalued and nonmonotone version of a nonlinear Navier–Fujita frictional slip condition. The analysis provides results on existence of solution to a variational–hemivariational inequality, continuous depen dence of the solution on the data, existence of solutions to optimal control problems, and the dependence of the solution on the yield limit. The proofs profit from results of nonsmooth analysis and the theory of multivalued pseudomontone operators