A unilateral contact 2D-problem is considered provided one of two elastic bodies can shift in a given direction as a rigid body. Using Lagrange multipliers for both normal and tangential constraints on the contact interface, we introduce a saddle point problem and prove its unique solvability. We discretize the problem by a standard finite element method and prove a convergence of approximations. We propose a numerical realization on the basis of an auxiliary “bolted” problem and the algorithm of...

The approximation of a mixed formulation of elliptic variational inequalities is studied. Mixed formulation is defined as the problem of finding a saddle-point of a properly chosen Lagrangian $\mathcal{2}$ on a certain convex set $Kx\Lambda$. Sufficient conditions, guaranteeing the convergence of approximate solutions are studied. Abstract results are applied to concrete examples.

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...

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...

This paper concerns an obstacle control problem for an elastic (homogeneous) and isotropic) pseudoplate. The state problem is modelled by a coercive variational inequality, where control variable enters the coefficients of the linear operator. Here, the role of control variable is played by the thickness of the pseudoplate which need not belong to the set of continuous functions. Since in general problems of control in coefficients have no optimal solution, a class of the extended optimal control...

Two-scale convergence is a powerful mathematical tool in periodic homogenization developed for modelling media with periodic structure. The contribution deals with the classical definition, its problems, the ``dual'' definition based on the so-called periodic unfolding. Since in the case of domains with boundary the unfolding operator introduced by D. Cioranescu, A. Damlamian, G. Griso does not satisfy the crucial integral preserving property, the contribution proposes a modified unfolding operator...

We propose a modification of the golden ratio algorithm for solving pseudomonotone equilibrium problems with a Lipschitz-type condition in Hilbert spaces. A new non-monotone stepsize rule is used in the method. Without such an additional condition, the theorem of weak convergence is proved. Furthermore, with strongly pseudomonotone condition, the $R$-linear convergence rate of the method is established. The results obtained are applied to a variational inequality problem, and the convergence rate...

We show that for every $\epsilon >0$, there is a set $A\subset {ℝ}^2$ such that ${\mathscr{H}}^1⌞A$ is a monotone measure, the corresponding tangent measures at the origin are not unique and ${\mathscr{H}}^1⌞A$ has the $1$-dimensional density between $1$ and $3+\epsilon$ everywhere on the support.

We consider the following classical autonomous variational problem$$\mathrm{minimize}\left\{F\left(v\right)={\int }_a^{b}f(v\left(x\right),v^{\text{'}}\left(x\right))x̣:v\in AC\left([a,b]\right),v\left(a\right)=\alpha ,v\left(b\right)=\beta \right\},$$where the Lagrangianf is possibly neither continuous, nor convex, nor coercive. We prove a monotonicity property of the minimizers stating that they satisfy the maximum principle or the minimum one. By virtue of such a property, applying recent results concerning constrained variational problems, we derive a relaxation theorem, the DuBois-Reymond necessary condition and some existence or non-existence criteria.

We consider the following classical autonomous variational problem$$\mathrm{minimize}\left\{F\left(v\right)={\int }_a^{b}f(v\left(x\right),v^{\text{'}}\left(x\right))x̣:v\in AC\left([a,b]\right),v\left(a\right)=\alpha ,v\left(b\right)=\beta \right\},$$ where the Lagrangian f is possibly neither continuous, nor convex, nor coercive. We prove a monotonicity property of the minimizers stating that they satisfy the maximum principle or the minimum one. By virtue of such a property, applying recent results concerning constrained variational problems, we derive a relaxation theorem, the DuBois-Reymond necessary condition and some existence or non-existence criteria.