Existence of a solution to the quasi-variational inequality problem arising in a modelfor sand surface evolution has been an open problem for a long time. Another long-standingopen problem concerns determining the dual variable, the flux of sand pouring down theevolving sand surface, which is also of practical interest in a variety of applications ofthis model. Previously, these problems were solved for the special case in which theinequality is simply variational. Here, we introduce a regularized mixed formulationinvolving both the primal (sand surface) and dual (sand flux) variables. We derive,analyse and compare two methods for the approximation, and numerical solution, of thismixed problem. We prove subsequence convergence of both approximations, as the meshdiscretization parameters tend to zero; and hence prove existence of a solution to thismixed model and the associated regularized quasi-variational inequality problem. One ofthese numerical approximations, in which the flux is approximated by thedivergence-conforming lowest order Raviart–Thomas element, leads to an efficient algorithmto compute not only the evolving pile surface, but also the flux of pouring sand. Resultsof our numerical experiments confirm the validity of the regularization employed.

We consider a class of quasi-variational inequalities arising in a large number of mathematical models, which describe quasi-static processes of contact between a deformable body and an obstacle, the so-called foundation. The novelty lies in the special structure of these inequalities that involve a history-dependent term as well as in the fact that the inequalities are formulated on the unbounded interval of time [0, +∞). We prove an existence and uniqueness result of the solution, then we complete it with a regularity result. The proofs are based on arguments of monotonicity and convexity, combined with a fixed point result obtained in [22]. We also describe a number of quasi-static frictional contact problems in which we model the material's behaviour with an elastic or viscoelastic constitutive law. The contact is modelled with normal compliance, with normal damped response or with the Signorini condition, as well, associated to versions of Coulomb's law of dry friction or to the frictionless condition. We prove that all these models cast in the abstract setting of history-dependent quasi-variational inequalities, with a convenient choice of spaces and operators. Then, we apply the abstract results in order to prove the unique weak solvability of each contact problem.

A zero-sum stochastic differential gameproblem on infinite horizon with continuous and impulse controls isstudied. We obtain the existence of the value of the game andcharacterize it as the unique viscosity solution of the associatedsystem of quasi-variational inequalities. We also obtain averification theorem which provides an optimal strategy of the game.