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

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

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.     

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.     

闭凸集上的一类非线性半变分不等式解的存在性

考虑一类定义在闭凸集上的非线性半变分不等式问题,通过运用闭凸集上的临界点理论、Clarke次微分性质以及非光滑紧性条件等,得到了这类半变分不等式解的存在性．     

Optimal Shape Problems for Eigenvalues

ABSTRACT

We consider the first Dirichlet eigenvalue for nonhomogeneous membranes. For given volume we want to find the domain which minimizes this eigenvalue. The problem is formulated as a variational free boundary problem. The optimal domain is characterized as the support of the first eigenfunction. We prove enough regularity for the eigenfunction to conclude that the optimal domain has finite parameter. Finally an overdetermined boundary value problem on the regular part of the free boundary is given.     

Continuity Results for a Class of Variational Inequalities with Applications to Time-Dependent Network Problems

We consider a class of monotone variational inequalities in which both the operator and the convex set are parametrized by continuous functions. Under suitable assumptions, we prove the continuity of the solution with respect to the parameter. As an important application, we consider the case of finite dimensional variational inequalities on suitable polyhedra. We demonstrate the applicability of our results to time dependent traffic equilibrium problem.     

On a Class of Random Variational Inequalities on Random Sets

We study a class of random variational inequalities on random sets and give measurability, existence, and uniqueness results in a Hilbert space setting. In the special case where the random and the deterministic variables are separated, we present a discretization technique based on averaging and truncation, prove a Mosco convergence result for the feasible random set, and establish norm convergence of the approximation procedure.     

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].     

Convergence of Rothe scheme for a class of dynamic variational inequalities involving Clarke subdifferential

In the first part of the paper we deal with a second-order evolution variational inequality involving a multivalued term generated by a Clarke subdifferential of a locally Lipschitz potential. For this problem we construct a time-semidiscrete approximation, known as the Rothe scheme. We study a sequence of solutions of the semidiscrete approximate problems and provide its weak convergence to a limit element that is a solution of the original problem. Next, we show that the solution is unique and the convergence is strong. In the second part of the paper, we consider a dynamic visco-elastic problem of contact mechanics. We assume that the contact process is governed by a normal damped response condition with a unilateral constraint and the body is non-clamped. The mechanical problem in its weak formulation reduces to a variational–hemivariational inequality that can be solved by finding a solution of a corresponding abstract problem related to one studied in the first part of the paper. Hence, we apply obtained existence result to provide the weak solvability of contact problem.     

一类非对称单调变分不等式的交替方向法

对一类非对称变分不等式问题提出了交替方向法。推广了交替方向仅适用于等式约束或不等约束的情形，得出了迭代序列的一些性质及收敛性．     

广义单调集值混合变分不等式的一类新算法

首先证明了广义单调集值混合变分不等式等价于一个新的不动点问题，在此基础上提出了解广义集值混合变分不等式及其相关优化问题的迭代算法，并给出了这类新算法的收敛性分析，我们的结果推广和综合了该领域的一些最新结论．     

Optimal Shape Design of Blazed Diffraction Gratings

The problem of designing a periodic interface between two materials in such a way that time-harmonic waves diffracted from the interface have a specified far-field pattern is studied. A minimization problem for the interface is formulated, and it is shown that solutions of constrained bounded variation exist. The differentiability of the cost functional is then studied, with no restrictions on the smoothness of the interface. Some computational issues are discussed, and finally the results of some numerical experiments are presented. Accepted 3 February 1998     

利用导数证明一类函数不等式

本文给出一个关于函数不等式的命题，用之证明了一类函数不等式．     

Optimal shape design for systems governed by variational inequalities,part 1: Existence theory for the elliptic case

Some general existence results for optimal shape design problems for systems governed by elliptic variational inequalities are established by the mapping method and variational convergence theory. Then, an existence theorem is given for the optimal shape for an electrochemical machining problem, in which the cost functional is not lower semicontinuous, by extending the general results to this case. Furthermore, this problem is approximated by a set of optimal shape design problems which have more smooth cost functionals and are easier to handle computationally.The authors with to express their sincere thanks to the reviewers for supplying additional references and for their valuable comments, which made the paper more readable.     

Optimal shape design for systems governed by variational inequalities,part 2: Existence theory for the evolution case

Some general existence results for optimal shape design problems for systems governed by parabolic variational inequalities are established by the mapping method and variational convergence theory. Then, an existence theorem is given for the optimal shape for an electrochemical machining problem, in which the cost functional is not lower semicontinuous, by extending the general results to this case. Furthermore, this problem is approximated by a set of optimal shape design problems which have more smooth cost functionals and are easier to handle computationally.The authors wish to express their sincere thanks to the reviewers for supplying additional references and for their valuable comments, which made the paper more readable.     

一类变分不等式问题的信赖域算法

基于J.M.Peng研究一类变分不等式问题（简记为VIP）时所提出的价值函数，本文提出了求解强单调的VIP的一个新的信赖域算法。和已有的处理VIP的信赖域方法不同的是：它在每步迭代时，不必求解带信赖域界的子问题，仅解一线性方程组而求得试验步。这样，计算的复杂性一般来说可降低。在通常的假设条件下，文中还证明了算法的整体收敛性。最后，在梯度是半光滑和约束是矩形域的假设下，该算法还是超线性收敛的。     

随机规划经验逼近问题最优解集的几乎处处下半收敛性

本文给出了随机规划经验逼近最优解集几乎处处下半收敛的一个充分条件,并由此得到随机规划经验逼近最优解集几乎处处Hausdorff收敛的一个充分条件.     

Existence Theorems for Systems of Generalized Vector Quasiequilibrium Problems and Optimization Problems

In this paper, we use existence theorems for the equilibria of generalized abstract economies proved recently to establish existence theorems for systems of generalized vector quasiequilibrium problems. Then, these existence theorems for equilibrium problems are used to derive existence theorems for systems of generalized vector quasivariational-like inequality problems, and vector quasioptimization problems.This research was supported by the National Science Council of the Republic of China. The authors express their gratitude to the referees for valuable suggestions.     

Iterative Algorithm for Finding Approximate Solutions of a Class of Mixed Variational-like Inequalities

The purpose of this paper is to investigate the iterative algorithm for finding approximate solutionsof a class of mixed variational-like inequalities in a real Hilbert space,where the iterative algorithm is presentedby virtue of the auxiliary principle technique.On one hand,the existence of approximate solutions of this classof mixed variational-like inequalities is proven.On the other hand,it is shown that the approximate solutionsconverge strongly to the exact solution of this class of mixed variational-like inequalities.