Unique Weak Solvability Research Articles

A two-dimensional elastic contact problem with unilateral constraints

We consider a mathematical model which describes the equilibrium of two elastic membranes fixed on their boundary and attached to an adhesive body, say a glue. The variational formulation of the model is in a form of an elliptic quasivariational inequality for the displacement field. We prove the unique weak solvability of the model, and then we state and prove a convergence result, for which we provide the corresponding mechanical interpretation. Next, we consider two associated optimization problems for which we provide existence results. Finally, we the present numerical simulation which validates our convergence result. We end this paper with some concluding remarks and an Appendix, in which we present the preliminary material needed in this paper.

On the unique weak solvability of second-order unconditionally stable difference scheme for the system of sine-Gordon equations

In the present paper, a nonlinear system of sine-Gordon equations that describes the DNA dynamics is considered. A novel unconditionally stable second-order accuracy difference scheme corresponding to the system of sine-Gordon equations is presented. In this work, for the first time in the literature, weak solution of this difference scheme is studied. The existence and uniqueness of the weak solution for the difference scheme are proved in the space of distributions, and the methods of variational calculus are applied. The finite-difference method and the fixed point theory are used in combination to perform numerical experiments that verify the theoretical statements.

Duality arguments in the analysis of a viscoelastic contact problem

We consider a mathematical model which describes the quasistatic frictionless contact of a viscoelastic body with a rigid-plastic foundation. We describe the mechanical assumptions, list the hypotheses on the data and provide three different variational formulations of the model in which the unknowns are the displacement field, the stress field and the strain field, respectively. These formulations have a different structure. Nevertheless, we prove that they are pairwise dual of each other. Then, we deduce the unique weak solvability of the contact problem as well as the Lipschitz continuity of its weak solution with respect to the data. The proofs are based on recent results on history-dependent variational inequalities and inclusions. Finally, we present numerical simulations in the study of the contact problem, together with the corresponding mechanical interpretations.

Modelling, analysis, and numerical simulation of a spring-rods system with unilateral constraints

In this paper, we consider a mathematical model which describes the equilibrium of two elastic rods attached to a nonlinear spring. We derive the variational formulation of the model which is in the form of an elliptic quasivariational inequality for the displacement field. We prove the unique weak solvability of the problem, then we state and prove some convergence results, for which we provide the corresponding mechanical interpretation. Next, we turn to the numerical approximation of the problem based on a finite element scheme. We use a relaxation method to solve the discrete problems that we implement on the computer. Using this method, we provide numerical simulations which validate our convergence results.

On boundary value problems for the Boussinesq-type equation with dynamic and non-dynamic boundary conditions

The work studies boundary value problems with non-dynamic and dynamic boundary conditions for one- and two-dimensional Boussinesq-type equations in domains representing a trapezoid, triangle, "curvilinear" trapezoid, "curvilinear" triangle, truncated cone, cone, truncated "curvilinear" cone, and "curvilinear" cone. Combining the methods of the theory of monotone operators and a priori estimates, in Sobolev classes, we have established theorems on the unique weak solvability of the boundary value problems under study.

Inverse problems for constrained parabolic variational-hemivariational inequalities *

In this paper we study a novel class of inverse problems for parabolic variational–hemivariational inequalities with a unilateral constraint. A theorem on the well-posedness for weak solution is established. Based on a new continuous dependence result, we prove the nonemptiness and stability of the set of optimal solutions to the corresponding inverse problems. We illustrate the results by a quasistatic nonsmooth frictional viscoelastic contact problem with a unilateral constraint for which we derive results on the unique weak solvability, and existence and stability of solutions to the associated inverse problems.

On a boundary value problem for a Boussinesq-type equation in a triangle

Earlier, we considered an initial-boundary value problem for a one-dimensional Boussinesq-type equation in a domain that is a trapezoid, in which the theorems on its unique weak solvability in Sobolev classes were established by the methods of the theory of monotone operators. In this article, we continue research in this direction and study the issues of correct formulation of the boundary value problem for a one-dimensional Boussinesq-type equation in a degenerate domain, which is a triangle. A scalar product is proposed with the help of which the monotonicity of the main operators is shown, and uniform a priori estimates are obtained. Further, using the methods of the theory of monotone operators and a priori estimates, theorems on its unique weak solvability in Sobolev classes are established. A theorem on increasing the smoothness of a weak solution is established. In proving the smoothness enhancement theorem, we use a generalization of the classical result on compactness in Banach spaces proved by Yu.I. Dubinsky ("Weak convergence in nonlinear elliptic and parabolic equations Sbornik: Mathematics, 67 (109): 4 (1965)) in the presence of a bounded set from a semi-normed space instead of a normed one. It is also shown that the solution may have a singularity at the point of degeneracy of the domain. The order of this feature is determined, and the corresponding theorem is proved.

Stochastic equations with time-dependent singular drift

We prove unique weak solvability for stochastic differential equations with drift in a large class of time-dependent vector fields. This class contains, in particular, the critical Ladyzhenskaya-Prodi-Serrin class, the weak Ld class as well as some vector fields that are not even in Lloc2+ε, ε>0.

An initial boundary value problem for the Boussinesq equation in a Trapezoid

This paper considers an initial boundary value problem for a one-dimensional Boussinesq-type equation in a domain, that is, a trapezoid. Using the methods of the theory of monotone operators, we establish theorems on their unique weak solvability in Sobolev classes.

A new class of history-dependent quasi variational–hemivariational inequalities with constraints

In this paper we consider an abstract class of time-dependent quasi variational–hemivariational inequalities which involves history-dependent operators and a set of unilateral constraints. First, we establish the existence and uniqueness of solution by using a recent result for elliptic variational–hemivariational inequalities in reflexive Banach spaces combined with a fixed-point principle for history-dependent operators. Then, we apply the abstract result to show the unique weak solvability to a quasistatic viscoelastic frictional contact problem. The contact law involves a unilateral Signorini-type condition for the normal velocity and the nonmonotone normal damped response condition while the friction condition is a version of the Coulomb law of dry friction in which the friction bound depends on the accumulated slip.

Dynamic viscoelastic unilateral constrained contact problems with thermal effects

A new model that describes a dynamic frictional contact between a viscoelastic body and an obstacle is investigated in this paper. We consider a nonlinear viscoelastic constitutive law which involves a convex subdifferential inclusion term and thermal effects. The contact condition is modeled with unilateral constraint condition for a version of normal velocity. The boundary conditions that describe the contact, friction and heat flux are govern by the generalized Clarke multivalued subdifferential. We derive a coupled system of two nonlinear first order evolution inclusions problems, which consists of a parabolic variational-hemivariational inequality for the displacement and a hemivariational inequality of parabolic type for the temperature. Then, the unique weak solvability of the contact problem is obtained by virtue of a fixed point theorem and the surjectivity result of multivalued maps. Finally, we deliver a continuous dependence result on a coupled system when the data are subjected to perturbations.

A unilateral constraint contact problem for viscoelastic materials with wear

The aim of this work is to investigate a mathematical model that describes the frictional contact between viscoelastic materials with a long memory and a moving foundation. The contact condition is modeled with a version of normal compliance condition with unilateral constraint in which wear of foundation is considered. Friction contact condition follows a sliding version of Coulomb's law of dry friction. We derive a variational formulation of the problem, which couples a variational inequality and an integral equation. Then, we obtain the unique weak solvability of the contact problem by virtue of a fixed point theorem. By using the penalization method, we discuss the convergence of the contact problem. Further, we obtain the unique solution of the penalized problem and this solution converges to the solution of the original contact model as the penalization parameter converges to zero.

A New Class of History–Dependent Evolutionary Variational–Hemivariational Inequalities with Unilateral Constraints

In this paper we study a new abstract evolutionary variational–hemivariational inequality which involves unilateral constraints and history–dependent operators. First, we prove the existence and uniqueness of solution by using a mixed equilibrium formulation with suitable selected functions together with a fixed-point principle for history–dependent operators. Then, we apply the abstract result to show the unique weak solvability to a dynamic viscoelastic frictional contact problem. The contact law involves a unilateral Signorini-type condition for the normal velocity combined with the nonmonotone normal damped response condition while the friction condition is a version of the Coulomb law of dry friction in which the friction bound depends on the accumulated slip.

Strong convergence of a linearization method for semi-linear elliptic equations with variable scaled production

This work is devoted to the development and analysis of a linearization algorithm for microscopic elliptic equations, with scaled degenerate production, posed in a perforated medium and constrained by the homogeneous Neumann–Dirichlet boundary conditions. This technique plays two roles: to guarantee the unique weak solvability of the microscopic problem and to provide a fine approximation in the macroscopic setting. The scheme systematically relies on the choice of a stabilization parameter in such a way as to guarantee the strong convergence in $$H^1$$ norm for both the microscopic and macroscopic problems. In the standard variational setting, we prove the $$H^1$$ -type contraction at the micro-scale based on the energy method. Meanwhile, we adopt the classical homogenization result in line with corrector estimate to show the convergence of the scheme at the macro-scale. In the numerical section, we use the standard finite element method to assess the efficiency and convergence of our proposed algorithm.

A Tykhonov-type well-posedness concept for elliptic hemivariational inequalities

In this paper, we introduce a new Tykhonov-type well-posedness concept for elliptic hemivariational inequalities, governed by an approximating function h. We characterize the well-posedness in terms of the metric properties of the family of approximating sets, under various assumptions on h. Then, we use the well-posedness properties in order to obtain convergence results of the solution with respect to the data. The proofs are based on arguments of monotonicity combined with the properties of the Clarke directional derivative. Our results provide mathematical tools in the study of a large number of static problems in Contact Mechanics. To provide an example, we consider a mathematical model which describes the equilibrium of a rod–spring system with unilateral constraints. We prove the unique weak solvability of the model, and then we illustrate our abstract convergence results in the study of this contact problem and provide the corresponding mechanical interpretations.

Fully history-dependent evolution hemivariational inequalities with constraints

In this paper we study a new class of abstract evolution first order hemivariational inequalities which involves constraints and history-dependent operators. First, we prove the existence and uniqueness of solution by using a mixed equilibrium formulation with suitable selected bifunctions combined with a fixed-point principle for history-dependent operators. Next, we deduce existence, uniqueness and regularity results for some special subclasses of problems which include a constrained history-dependent variational–hemivariational inequality, an evolution quasi-variational inequality with constraints, and an evolution second order hemivariational inequality with constraints. Then, we provide an application of the results to a dynamic unilateral viscoelastic frictional contact problem and show its unique weak solvability.

A new class of quasistatic frictional contact problems governed by a variational–hemivariational inequality

A quasistatic nonsmooth frictional contact problem for a viscoelastic material is studied. The contact is modeled by a multivalued normal damped response condition with the Clarke generalized gradient of a locally Lipschitz superpotential and the friction is described by a version of the Coulomb law of dry friction with the friction bound depending on the regularized normal stress. The weak formulation of the contact problem is a history-dependent variational–hemivariational inequality coupled with an operator equation for the stress field. A result on the unique weak solvability to this coupled system is proved through a recent contribution on history-dependent subdifferential inclusions and a fixed point approach.

Boundary optimal control of a nonsmooth frictionless contact problem

We consider a mathematical model which describes the equilibrium of an elastic body in contact with a foundation. The contact is frictionless and is modeled with a nonsmooth interface law which involves unilateral constraints and subdifferential conditions. The weak formulation of the model is in the form of an elliptic hemivariational inequality governed by a number of parameters. We prove the unique weak solvability of the problem, then we state and prove a continuous dependence result of the solution with respect to the data and parameters. Next, we formulate a boundary optimal control problem for which we prove the existence of optimal pairs. We also study the dependence of the optimal pairs with respect to the data and parameters and prove a convergence result. The proofs are based on arguments of monotonicity, lower semicontinuity and Clarke subdifferential calculus.

Time-dependent inclusions and sweeping processes in contact mechanics

We consider a class of time-dependent inclusions in Hilbert spaces for which we state and prove an existence and uniqueness result. The proof is based on arguments of variational inequalities, convex analysis and fixed point theory. Then we use this result to prove the unique weak solvability of a new class of Moreau's sweeping processes with constraints in velocity. Our results are useful in the study of mathematical models which describe the quasistatic evolution of deformable bodies in contact with an obstacle. To provide some examples we consider three viscoelastic contact problems which lead to time-dependent inclusions and sweeping processes in which the unknowns are the displacement and the velocity fields, respectively. Then we apply our abstract results in order to prove the unique weak solvability of the corresponding contact problems.

Well-posedness for a class of frictional contact models via mixed variational formulations

We analyze a class of frictional contact problems by means of a Lagrange multipliers technique. The mathematical model consists of a boundary value problem governed by the p-Laplace operator. Firstly, we deliver a mixed variational formulation. Secondly, we discuss the unique solvability of the mixed variational problem into an abstract functional framework, paying attention to the stability of the solution, too. Then, we apply the abstract results in order to study the unique weak solvability of the boundary value problem under consideration. The stability of the weak solution is also discussed.