We show that the spectrum of the relativistic mean curvature operator on a bounded domain Ω ⊂ ℝN (N ⩾ 1) having smooth boundary, subject to the homogeneous Dirichlet boundary condition, is exactly the interval (λ1(2), ∞), where λ1(2) stands for the principal frequency of the Laplace operator in Ω.

We study the numerical approximation of doubly reflected backward stochastic differentialequations with intermittent upper barrier (RIBSDEs). These denote reflected BSDEs in whichthe upper barrier is only active on certain random time intervals. From the point of viewof financial interpretation, RIBSDEs arise as pricing equations of game options withconstrained callability. In a Markovian set-up we prove a convergence rate for atime-discretization scheme by simulation to an RIBSDE. We also characterize the solutionof an RIBSDE as the largest viscosity subsolution of a related system of variationalinequalities, and we establish the convergence of a deterministic numerical scheme forthat problem. Due to the potentially very high dimension of the system of variationalinequalities, this approach is not always practical. We thus subsequently prove aconvergence rate for a time-discretisation scheme by simulation to an RIBSDE.

We are concerned with the proof of a generalized solution to an ill-posed variational inequality. This is determined as a solution to an appropriate minimization problem involving a nonconvex functional, treated by an optimal control technique.

In the article an optimal control problem subject to a stationary variational inequalityis investigated. The optimal control problem is complemented with pointwise controlconstraints. The convergence of a smoothing scheme is analyzed. There, the variationalinequality is replaced by a semilinear elliptic equation. It is shown that solutions ofthe regularized optimal control problem converge to solutions of the original one. Passingto the limit in the optimality system of the regularized problem allows to proveC-stationarity of local solutions of the original problem. Moreover, convergence rateswith respect to the regularization parameter for the error in the control are obtained,which turn out to be sharp. These rates coincide with rates obtained by numericalexperiments, which are included in the paper.

In this paper sufficient second order optimality conditions for optimal control problems subject to stationary variational inequalities of obstacle type are derived. Since optimality conditions for such problems always involve measures as Lagrange multipliers, which impede the use of efficient Newton type methods, a family of regularized problems is introduced. Second order sufficient optimality conditions are derived for the regularized problems as well. It is further shown that these conditions are also sufficient for superlinear convergence of the semi-smooth Newton algorithm to be well-defined and superlinearly convergent when applied to the first order optimality system associated with the regularized problems.

Caused by the problem of unilateral contact during vibrations of satellite solar arrays, the aim of this paper is to better understand such a phenomenon. Therefore, it is studied here a simplified model composed by a beam moving between rigid obstacles. Our purpose is to describe and compare some families of fully discretized approximations and their properties, in the case of non-penetration Signorini's conditions. For this, starting from the works of Dumont and Paoli, we adapt to our beam model the singular dynamic method introduced by Renard. A particular emphasis is given in the use of a restitution coefficient in the impact law. Finally, various numerical results are presented and energy conservation capabilities of the schemes are investigated.

The contact between two membranes can be described by a system of variationalinequalities, where the unknowns are the displacements of the membranes and theaction of a membrane on the other one. We first perform the analysis of thissystem. We then propose a discretization, where the displacements areapproximated by standard finite elements and the action by alocal postprocessing. Such a discretization admits an equivalent mixedreformulation. We prove the well-posedness of the discrete problem and establishoptimal a priori error estimates.

We consider mixed and hybrid variational formulations to the linearizedelasticity system in domains with cracks. Inequality type conditions areprescribed at the crack faces which results in unilateral contact problems. Thevariational formulations are extended to the whole domain including the crackswhich yields, for each problem, a smooth domain formulation. Mixedfinite element methods such as PEERS or BDM methods are designed to avoidlocking for nearly incompressible materials in plane elasticity. We study andimplement discretizations based on such mixed finite element methods for thesmooth domain formulations to the unilateral crack problems. We obtainconvergence rates and optimal error estimates and we present some numericalexperiments in agreement with the theoretical results.

From the fundamental laws of elasticity, we write a model for the contact between two membranes and we perform the analysis of the corresponding system of variational inequalities. We propose a finite element discretization of this problem and prove its well-posedness. We also establish a priori and a posteriori error estimates.

In this paper, we introduce and consider a new class of complementarity problems, which are called the generalized mixed quasi-complementarity problems in a topological vector space. We show that the generalized mixed quasi-complementarity problems are equivalent to the generalized mixed quasi variational inequalities. Using a new type of KKM mapping theorem, we study the existence of a solution of the generalized mixed quasi-variational inequalities and generalized mixed quasi-complementarity problems. Several special cases are also discussed. The results obtained in this paper can be viewed as extension and generalization of the previously known results.

For a large class of operator inclusions, including those generated by maps of pseudomonotone type, we obtain a general theorem on existence of solutions. We apply this result to some particular examples. This theorem is proved using the method of difference approximations.

The numerical modeling of the fully developed Poiseuille flow of a Newtonian fluid in a square section with slip yield boundary condition at the wall is presented. The stick regions in outer corners and the slip region in the center of the pipe faces are exhibited. Numerical computations cover the complete range of the dimensionless number describing the slip yield effect, from a full slip to a full stick flow regime. The resolution of variational inequalities describing the flow is based on the augmented Lagrangian method and a finite element method. The localization of the stick-slip transition points is approximated by an anisotropic auto-adaptive mesh procedure. The singular behavior of the solution at the neighborhood of the stick-slip transition point is investigated.

A general setting is proposed for the mixed finite element approximations of elliptic differential problems involving a unilateral boundary condition. The treatment covers the Signorini problem as well as the unilateral contact problem with or without friction. Existence, uniqueness for both the continuous and the discrete problem as well as error estimates are established in a general framework. As an application, the approximation of the Signorini problem by the lowest order mixed finite element method of Raviart–Thomas is proved to converge with a quasi-optimal error bound.

We propose a variational model for one of the most importantproblems in traffic networks, namely, the network equilibrium flow that is, traditionallyin the context of operations research, characterized by minimum cost flow. This model has the peculiarity of being formulated by means of a suitable variational inequality (VI) andits solution is called “equilibrium”. This model becomes a minimum cost model when the cost function is separableor, more general, when the Jacobian of the cost operator is symmetric;in such cases a functional representing the total network utility exists.In fact in these cases we can write the first order optimality conditions which turn out to be a VI.In the other situations (i.e., when global utility functional does not exist),which occur much more often in the real problems, we can study the network by looking for equilibrium solutions instead of minimum points.The Lagrangean approach to the study of the VI allows us to introduce dual variables, associated to the constraints of the feasible set, which may receive interesting interpretations in terms of potentials associated to the arcs and the nodes of the network.This interpretation is an extension and generalization of the classic Bellman conditions. Finally, we deepen the analysis of the networks having capacity constraints.


Semi–smooth Newton methods are analyzed for a class of variational inequalities in infinite dimensions.It is shown that they are equivalent to certain active set strategies. Global and local super-linear convergence areproved. To overcome the phenomenon of finite speed of propagation of discretized problems a penalty versionis used as the basis for a continuation procedure to speed up convergence. The choice of the penalty parameter can be made on the basis of an L∞ estimate for the penalized solutions. Unilateral as well as bilateral problems are considered.

On expose les difficultés d'ordremathématique que posent des modèles récents de sédimentation-érosion de bassins élaborés par l'InstitutFrançais du Pétrole et fondés sur la prise en compte dediverses contraintes d'unilatéralité. On présente quelquesrésultats partiels théoriques et des directions de recherche pour larésolution d'un problème inverse posé par l'étudestratigraphique d'une colonne monolithologique.

We study first order optimality systems for the control of a systemgoverned by avariationalinequality and deal with Lagrange multipliers: isit possible to associate to each pointwise constraint a multiplier to get a “good” optimality system? We givepositive and negative answers for the finite and infinite dimensional cases.These results are compared withthe previous ones got by penalization or differentiation.