Dynamic Hemivariational Inequalities and Their Applications

Dynamic hemivariational inequalities are studied in the present paper. Starting from their solution in the distributional sense, we give certain existence and approximation results by using the Faedo–Galerkin method and certain compactness arguments. Applications from mechanics (viscoelasticity) illustrate the theory.     

An Existence Theorem for Wave‐Type Hyperbolic Hemivariational Inequalities

In this paper we prove the existence of solutions for a hyperbolic hemivariationalinequality of the form u″ + Bu + ∂j (u) ∋ f where B is a linear elliptic operator and ∂j is the Clarke subdifferential of a locally Lipschitz function j. Our result is based on the parabolic regularization method.     

Some Convergence Results for Evolution Hemivariational Inequalities

This paper is devoted to the regularization of a class of evolution hemivariational inequalities. The operator involved is taken to be non-coercive and the data are assumed to be known approximately. Under the assumption that the evolution hemivariational inequality be solvable, a weakly convergent approximation procedure is designed by means of the so-called Browder-Tikhonov regularization method.     

拟线性椭圆型H-半变分不等式

本文研究一类拟线性椭圆型H-半变分不等式,即研究具有非凸、非光滑泛函的椭圆型不等式。这类问题的研究来自力学。利用Clarke广义梯度和伪单调算子理论,我们证明了拟线性椭圆型H-半变分不等式解的存在性。     

发展型H—半变分不等式解的存在性

本文研究非线性发展型H－半变分不等式,即具有非凸泛函的抛物型变分不等式,这类问题的研究起源于力学。利用Clarke广义梯度和（S＋）型多值映象的不动点理论,我们证明了这类问题解的存在性。并利用这一理论,研究了具间断项的非线性抛物型方程解的存在性。     

一类抛物型H-半变分不等式

研究一类拟线性抛物型H-半变分不等式,即研究具有非凸、非光滑泛函的抛物型变分不等式。这类问题的研究来自力学。利用Clarke广义梯度和伪单调算子理论,证明了一类拟线性抛物型H-半变分不等式解的存在性。     

Boundary Stabilization of Hyperbolic Hemivariational Inequalities

In this paper, we prove the existence of weak solutions and investigate uniform decay rates of global weak solutions for a hyperbolic hemivariational inequalitiy of dynamic elasticity. This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-355-C00002).     

Existence Results for Evolution Noncoercive Hemivariational Inequalities

This paper is devoted to the existence of solutions for evolution hemivariational inequalities as generalizations of evolution variational inequalities to nonconvex functionals. The operators involved are taken to be multivalued and noncoercive. Using the notion of the generalized gradient of Clarke and the recession method, some existence results of solutions are proved.     

Regularized Equilibrium Problems with Application to Noncoercive Hemivariational Inequalities

In this paper, we are concerned with noncoercive equilibrium problems associated with a bifunction which does not satisfy necessarily an algebraic monotonicity assumption. Our tool is a regularization procedure which we develop for equilibrium problems. The abstract existence result established is then applied to the solution of noncoercive hemivariational inequalities.     

Optimal Shape Design Problems for a Class of Systems Described by Hemivariational Inequalities

Optimal shape design problems for systems governed by an elliptic hemivariational inequality are considered. A general existence result for this problem is established by the mapping method.     

Perturbations of Hemivariational Inequalities with Constraints and Applications

The aim of the present paper is to discuss the influence which certain perturbations have on the solution of the eigenvalue problem for hemivariational inequalities on a sphere of given radius. The perturbation results in adding a term of the type >g ⁰(x, u(x); v(x)) to the hemivariational inequality, where g is a locally Lipschitz nonsmooth and nonconvex energy functional. Applications illustrate the theory.     

Nontrivial Solutions for Resonant Hemivariational Inequalities

We study a resonant semilinear elliptic hemivariational inequality. Under some assumptions of strong resonance on the Clarke subdifferential of the superpotential, and using nonsmooth critical point theory, the existence of a nontrivial solution of the problem is shown. This paper has been partially supported by the State Committee for Scientific Research of Poland (KBN) under research grants no. 2 P03A 003 25 and no. 4 T07A 027 26.     

Multiplicity Results for an Eigenvalue Problem for Hemivariational Inequalities in Strip-Like Domains

In this paper we study the multiplicity of solutions for a class of eigenvalue problems for hemivariational inequalities in strip-like domains. The first result is based on a recent abstract theorem of Marano and Motreanu, obtaining at least three distinct, axially symmetric solutions for certain eigenvalues. In the second result, a version of the fountain theorem of Bartsch which involves the nonsmooth Cerami compactness condition, provides not only infinitely many axially symmetric solutions but also axially nonsymmetric solutions in certain dimensions. In both cases the principle of symmetric criticality for locally Lipschitz functions plays a crucial role.Mathematics Subject Classifications (2000) 35A15, 35P30, 35J65.Supported by the EU Research Training Network HPRN-CT-1999-00118.     

On the Approximate Controllability for Hilfer Fractional Evolution Hemivariational Inequalities

The paper is concerned with the approximate controllability of some Hilfer fractional evolution hemivariational inequalities. Using two classes of operators and their fundamental properties, we derive sufficient conditions for approximate controllability of linear and semilinear controlled systems via a fixed point theorem for multivalued maps. Finally, an example is given to illustrate our theory.     

Finite Element Approximation of Vector-Valued Hemivariational Problems

In this paper we develop a finite element approximationfor vector-valuedhemivariational inequalities.This class of hemivariational problems wasintroducedin [12],[13]. We study two differentproblems: unconstrained oneand constrained one witha nonempty, closed, convex constraint set K.We shall show firstly that the discrete problemsare solvable by usingconsequences of Kakutanifixed point theorem and secondly that the solutionsof the discrete problemsare close on subsequences to the continuous ones.     

Finite Element Approximation of Hemivariational Inequalities

The paper deals with a finite element approximation of elliptic and parabolic variational inequalities. Elliptic hemivariational inequalities are equivalently expressed as a system consisting of one equation and one inclusion for a couple of unknowns, namely a primal variable u and an element belonging to a multivalued mapping at u. Both components of the solution are approximated independently each other by a finite element method. Parabolic inequalities are transformed into a system of elliptic ones by using an appropriate time discretization. A numerical experiment is realized by using non-smooth optimization methods.     

Inverse Coefficient Problems for Elliptic Hemivariational Inequalities

This paper is devoted to a class of inverse coefficient problems for nonlinear elliptic hemivariational inequalities. The unknown coefficient of elliptic hemivariational inequalities depends on the gradient of the solution and belongs to a set of admissible coefficients. It is shown that the nonlinear elliptic hemivariational inequalities are uniquely solvable for the given class of coefficients. The result of existence of quasisolutions of the inverse problems is obtained.     

A Nontrivial Solution for Neumann Noncoercive Hemivariational Inequalities

In this paper we consider Neumann noncoercive hemivariational inequalities, focusing on nontrivial solutions. We use the critical point theory for locally Lipschitz functionals.     

On the Existence of Positive Solutions for Hemivariational Inequalities Driven by the p-Laplacian

We study nonlinear elliptic problems driven by the p-Laplacian and with a nonsmooth locally Lipschitz potential (hemivariational inequality). We do not assume that the nonsmooth potential satisfies the Ambrosetti--Rabinowitz condition. Using a variational approach based on the nonsmooth critical point theory, we establish the existence of at least one smooth positive solution.Mathematics Subject Classifications (2000). 35J50, 35J85, 35R70.This article is Revised version.Leszek Gasiski is an award holder of the NATO Science FellowshipProgramme, which was spent in the National Technical University of Athens.     

On the numerical treatment of nonconvex energy problems of mechanics

The present paper presents three numerical methods devised for the solution of hemivariational inequality problems. The theory of hemivariational inequalities appeared as a development of variational inequalities, namely an extension foregoing the assumption of convexity that is essentially connected to the latter. The methods that follow partly constitute extensions of methods applied for the numerical solution of variational inequalities. All three of them actually use the solution of a central convex subproblem as their kernel. The use of well established techniques for the solution of the convex subproblems makes up an effective, reliable and versatile family of numerical algorithms for large scale problems. The first one is based on the decomposition of the contigent cone of the (super)-potential of the problem into convex components. The second one uses an iterative scheme in order to approximate the hemivariational inequality problem with a sequence of variational inequality problems. The third one is based on the fact that nonconvexity in mechanics is closely related to irreversible effects that affect the Hessian matrix of the respective (super)-potential. All three methods are applied to solve the same problem and the obtained results are compared.