Boundary Hemivariational Inequalities of Hyperbolic Type and Applications

In this paper we examine two classes of nonlinear hyperbolic initial boundary value problems with nonmonotone multivalued boundary conditions characterized by the Clarke subdifferential. We prove two existence results for multidimensional hemivariational inequalities: one for the inequalities with relation between reaction and velocity and the other for the expressions containing the reaction–displacement law. The existence of weak solutions is established by using a surjectivity result for pseudomonotone operators and a priori estimates. We present also an example of dynamic viscoelastic contact problem in mechanics which illustrate the applicability of our results.Mathematics Subject Classifications (2000). 34G20, 35A15, 35L85, 35L70, 74H20     

A class of dynamic frictional contact problems governed by a system of hemivariational inequalities in thermoviscoelasticity

In this paper we prove the existence and uniqueness of the weak solution for a dynamic thermoviscoelastic problem which describes frictional contact between a body and a foundation. We employ the nonlinear constitutive viscoelastic law with a long-term memory, which includes the thermal effects and considers the general nonmonotone and multivalued subdifferential boundary conditions for the contact, friction and heat flux. The model consists of the system of the hemivariational inequality of hyperbolic type for the displacement and the parabolic hemivariational inequality for the temperature. The existence of solutions is proved by using recent results from the theory of hemivariational inequalities and a fixed point argument.     

A class of optimal control problems for piezoelectric frictional contact models

We consider control problems for a mathematical model describing the frictional bilateral contact between a piezoelectric body and a foundation. The material’s behavior is modeled with a linear electro–elastic constitutive law, the process is static and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity conditions on the contact surface are described with the Clarke subdifferential boundary conditions. The weak formulation of the problem consists of a system of two hemivariational inequalities. We provide the results on existence and uniqueness of a weak solution to the model and, under additional assumptions, the continuous dependence of a solution on the data. Finally, for a class of optimal control problems and inverse problems, we prove the existence of optimal solutions.     

Existence results for evolutionary inclusions and variational–hemivariational inequalities

We consider an abstract first-order evolutionary inclusion in a reflexive Banach space. The inclusion contains the sum of L-pseudomonotone operator and a maximal monotone operator. We provide an existence theorem which is a generalization of former results known in the literature. Next, we apply our result to the case of nonlinear variational–hemivariational inequalities considered in the setting of an evolution triple of spaces. We specify the multivalued operators in the problem and obtain existence results for several classes of variational–hemivariational inequality problems. Finally, we illustrate our existence result and treat a class of quasilinear parabolic problems under nonmonotone and multivalued flux boundary conditions.     

History-dependent subdifferential inclusions and hemivariational inequalities in contact mechanics

We consider a class of subdifferential inclusions involving a history-dependent term for which we provide an existence and uniqueness result. The proof is based on arguments on pseudomonotone operators and fixed point. Then we specialize this result in the study of a class of history-dependent hemivariational inequalities. Such kind of problems arises in a large number of mathematical models which describe quasistatic processes of contact between a deformable body and an obstacle, the so-called foundation. To provide an example we consider a viscoelastic problem in which the frictional contact is modeled with subdifferential boundary conditions. We prove that this problem leads to a history-dependent hemivariational inequality in which the unknown is the velocity field. Then we apply our abstract result in order to prove the unique weak solvability of the corresponding contact problem.     

Analysis of a quasistatic contact problem for piezoelectric materials

We consider a mathematical model which describes the frictional contact between a piezoelectric body and an electrically conductive foundation. The process is quasistatic, the material behavior is modeled with an electro-viscoelastic constitutive law and the contact is described with subdifferential boundary conditions. We derive the variational formulation of the problem which is in the form of a system involving two history-dependent hemivariational inequalities in which the unknowns are the velocity and electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on a recent result on history-dependent hemivariational inequalities obtained in Migórski et al. (submitted for publication) [16].     

Subdifferential inclusions and quasi-static hemivariational inequalities for frictional viscoelastic contact problems

We survey recent results on the mathematical modeling of nonconvex and nonsmooth contact problems arising in mechanics and engineering. The approach to such problems is based on the notions of an operator subdifferential inclusion and a hemivariational inequality, and focuses on three aspects. First we report on results on the existence and uniqueness of solutions to subdifferential inclusions. Then we discuss two classes of quasi-static hemivariational ineqaulities, and finally, we present ideas leading to inequality problems with multivalued and nonmonotone boundary conditions encountered in mechanics.     

History-dependent variational–hemivariational inequalities in contact mechanics

We consider an abstract class of variational–hemivariational inequalities which arise in the study of a large number of mathematical models of contact. The novelty consists in the structure of the inequalities which involve two history-dependent operators and two nondifferentiable functionals, a convex and a nonconvex one. For these inequalities we provide an existence and uniqueness result of the solution. The proof is based on arguments of surjectivity for pseudomonotone operators and fixed point. Then, we consider a viscoelastic problem in which the contact is frictionless and is modeled with a new boundary condition which describes both the instantaneous and the memory effects of the foundation. We prove that this problem leads to a history-dependent variational–hemivariational inequality in which the unknown is the displacement field. We apply our abstract result in order to prove the unique weak solvability of this viscoelastic contact problem.     

Lagrange Multipliers and mixed formulations for some inequality constrained variational inequalities and some nonsmooth unilateral problems

Abstract

This paper addresses a class of inequality constrained variational inequalities and nonsmooth unilateral variational problems. We present mixed formulations arising from Lagrange multipliers. First we treat in a reflexive Banach space setting the canonical case of a variational inequality that has as essential ingredients a bilinear form and a non-differentiable sublinear, hence convex functional and linear inequality constraints defined by a convex cone. We extend the famous Brezzi splitting theorem that originally covers saddle point problems with equality constraints, only, to these nonsmooth problems and obtain independent Lagrange multipliers in the subdifferential of the convex functional and in the ordering cone of the inequality constraints. For illustration of the theory we provide and investigate an example of a scalar nonsmooth boundary value problem that models frictional unilateral contact problems in linear elastostatics. Finally we discuss how this approach to mixed formulations can be further extended to variational problems with nonlinear operators and equilibrium problems, and moreover, to hemivariational inequalities.     

EXISTENCE OF PERIODIC SOLUTIONS FOR A DIFFERENTIAL INCLUSION SYSTEMS INVOLVING THE p（t）-LAPLACIAN

We study a nonlinear periodic problem driven by the p(t)-Laplacian and having a nonsmooth potential (hemivariational inequalities). Using a variational method based on nonsmooth critical point theory for locally Lipschitz functions, we first prove the existence of at least two nontrivial solutions under the generalized subquadratic and then establish the existence of at least one nontrivial solution under the generalized superquadratic.     

Analysis of an elasto-piezoelectric system of hemivariational inequalities with thermal effects

In this paper we prove the existence and uniqueness of a weak solution for a dynamic electo-viscoelastic problem that describes a contact between a body and a foundation. We assume the body is made from thermoviscoelastic material and consider nonmonotone boundary conditions for the contact. We use recent results from the theory of hemivariational inequalities and the fixed point theory.     

On convergence of numerical methods for variational–hemivariational inequalities under minimal solution regularity

Hemivariational inequalities have been successfully employed for mathematical and numerical studies of application problems involving nonsmooth, nonmonotone and multivalued relations. In recent years, error estimates have been derived for numerical solutions of hemivariational inequalities under additional solution regularity assumptions. Since the solution regularity properties have not been rigorously proved for hemivariational inequalities, it is important to explore the convergence of numerical solutions of hemivariational inequalities without assuming additional solution regularity. In this paper, we present a general convergence result enhancing existing results in the literature.     

Existence of antiperiodic solutions for hemivariational inequalities

In this paper we consider the problem of existence of antiperiodic solutions for first-order and second-order hemivariational inequalities with a pseudomonotone operator. We first give the surjectivity result and then prove a existence of antiperiodic solutions for hemivariational inequalities with the surjectivity result.     

Three solutions for an obstacle problem for a class of variational–hemivariational inequalities

In this paper, we consider a class of obstacle problems for variational–hemivariational inequalities, by using nonsmooth version of three points critical theory in [S.A. Marano, D. Motreanu, On a three critical points theorem for non-differentiable functions and application to nonlinear boundary value problems, Nonlinear Anal. 48 (2002) 37–52], the existence of three solutions for the problem is obtained.     

A class of hemivariational inequalities for nonstationary Navier–Stokes equations

This paper is devoted to the study of a class of hemivariational inequalities for the time-dependent Navier–Stokes equations, including both boundary hemivariational inequalities and domain hemivariational inequalities. The hemivariational inequalities are analyzed in the framework of an abstract hemivariational inequality. Solution existence for the abstract hemivariational inequality is explored through a limiting procedure for a temporally semi-discrete scheme based on the backward Euler difference of the time derivative, known as the Rothe method. It is shown that solutions of the Rothe scheme exist, they contain a weakly convergent subsequence as the time step-size approaches zero, and any weak limit of the solution sequence is a solution of the abstract hemivariational inequality. It is further shown that under certain conditions, a solution of the abstract hemivariational inequality is unique and the solution of the abstract hemivariational inequality depends continuously on the problem data. The results on the abstract hemivariational inequality are applied to hemivariational inequalities associated with the time-dependent Navier–Stokes equations.     

Dynamic thermoviscoelastic problem with friction and damage

In this paper we present a model of dynamic frictional contact between a thermoviscoelastic body and a foundation. The thermoviscoelastic constitutive law includes a temperature effect described by the parabolic equation with the subdifferential boundary condition and a damage effect described by the parabolic inclusion with the homogeneous Neumann boundary condition. Contact is modeled with bilateral condition and is associated to a subdifferential frictional law. The variational formulation of the problem leads to a system of hyperbolic hemivariational inequality for the displacement, parabolic hemivariational inequality for the temperature and parabolic variational inequality for the damage. The existence of a unique weak solution is proved by using recent results from the theory of hemivariational inequalities, variational inequalities, and a fixed point argument.     

On the approximate controllability for fractional evolution hemivariational inequalities

In this paper, we focus on the approximate controllability of control systems described by a large class of fractional evolution hemivariational inequalities. Firstly, we introduce the concept of mild solutions and present the existence of mild solutions for this kind of problems. Next, we show the approximate controllability of the corresponding linear control system. Finally, the approximate controllability of the fractional evolution hemivariational inequalities is formulated and proved under some appropriate conditions. An example demonstrates the applicability of our results. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence of anti-periodic solutions for hemivariational inequalities

J. Y. Park and T. G. Ha [Nonlinear Anal., 2008, 68: 747-767; 2009, 71: 3203-3217] investigated the existence of anti-periodic solutions for hemivariational inequalities with a pseudomonotone operator. In this note, we point out that the methods used there are not suitable for the proof of the existence of anti-periodic solutions for hemivariational inequalities and we shall give a straightforward approach to handle these problems. The main tools in our study are the maximal monotone property of the derivative operator with antiperiodic conditions and the surjectivity result for L-pseudomonotone operators.     

Hemivariational inequality approach to the dynamic viscoelastic contact problem with nonmonotone normal compliance and slip-dependent friction

We examine a mathematical model which describes dynamic viscoelastic contact problems with nonmonotone normal compliance condition and the slip displacement dependent friction. First, we derive a weak formulation of the model in the form of a hemivariational inequality. Then we embed the hemivariational inequality into a class of second-order evolution inclusions for which we provide a result on the existence of a solution. We conclude with examples of the subdifferential boundary conditions for contact with normal compliance and the slip dependent friction.     

From solvability and approximation of variational inequalities to solution of nondifferentiable optimization problems in contact mechanics

In this paper, we first gather existence results for linear and for pseudo-monotone variational inequalities in reflexive Banach spaces. We discuss the necessity of the involved coerciveness conditions and their relationship. Then, we combine Mosco convergence of convex closed sets with an approximation of pseudo-monotone bifunctions and provide a convergent approximation procedure for pseudo-monotone variational inequalities in reflexive Banach spaces. Since hemivariational inequalities in linear elasticity are pseudo-monotone, our approximation method applies to nonmonotone contact problems. We sketch how regularization of the involved nonsmooth functionals together with finite element approximation lead to an efficient numerical solution method for these nonconvex nondifferentiable optimization problems. To illustrate our theory, we give a numerical example of a 2D linear elastic block under a given nonmonotone contact law.