Search results for "Variational inequality"

Overview of Other Results and Open Problems

2014

This chapter presents an overview of results related to error control methods, which were not considered in previous chapters. In the first part, we discuss possible extensions of the theory exposed in Chaps. 3 and 4 to nonconforming approximations and certain classes of nonlinear problems. Also, we shortly discuss some results related to explicit evaluation of modeling errors. The remaining part of the chapter is devoted to a posteriori estimates of errors in iteration methods. Certainly, the overview is not complete. A posteriori error estimation methods are far from having been fully explored and this subject contains many unsolved problems and open questions, some of which we formulate …

Regularity of solutions of nonlinear variational inequalities

1973

On State Constrained Optimal Shape Design Problems

1987

This paper is concerned with the following optimal design problem with constraints both on the state and on the control: $$MinimizeJ(y,u)$$ (P) subject to $$A\left( u \right)y + \partial \varphi \left( y \right) \mathrel\backepsilon Bu + f,$$ (1.1) $$y \in K,$$ (1.2) $$u \in {U_{ad}}.$$ (1.3)

Contact Shape Optimization

1995

Shape optimization is a branch of the optimal control theory in which the control variable is connected with the geometry of the problem. The aim is to find a shape from an a priori defined class of domains, for wich the corresponding cost functional attains its minimum. Shape optimization of mechanical systems, behaviour of which is described by equations, has been very well analyzed from the mathematical, as well as from the mechanical point of view, see [1], [2], [3] and references therein. The aim of this contribution is to extend results to the case, in which the system is described by the so called variational inequalities. There are two reasons for doing that: 1) The behavior of many…

Optimal Control for Plate Problems

2003

The variational approach leading to indirect methods Optimal Control Problems is applied to the study of simply supported and clamped plates. A unified approach based on distributed optimal control problems governed by second order elliptic boundary value problems and their penalization is used.

phi-Best proximity point theorems and applications to variational inequality problems

2017

The main concern of this study is to introduce the notion of $$\varphi $$ -best proximity points and establish the existence and uniqueness of $$\varphi $$ -best proximity point for non-self mappings satisfying $$(F,\varphi )$$ -proximal and $$(F,\varphi )$$ -weak proximal contraction conditions in the context of complete metric spaces. Some examples are supplied to support the usability of our results. As applications of the obtained results, some new best proximity point results in partial metric spaces are presented. Furthermore, sufficient conditions to ensure the existence of a unique solution for a variational inequality problem are also discussed.

Existence and Relaxation Results for Second Order Multivalued Systems

2021

AbstractWe consider nonlinear systems driven by a general nonhomogeneous differential operator with various types of boundary conditions and with a reaction in which we have the combined effects of a maximal monotone term $A(x)$ A ( x ) and of a multivalued perturbation $F(t,x,y)$ F ( t , x , y ) which can be convex or nonconvex valued. We consider the cases where $D(A)\neq \mathbb{R}^{N}$ D ( A ) ≠ R N and $D(A)= \mathbb{R}^{N}$ D ( A ) = R N and prove existence and relaxation theorems. Applications to differential variational inequalities and control systems are discussed.

Steady-state radiation heat transfer problem

1996

In Section 8.2, we shall see that the steady-state radiative heat transfer problem can be transformed to minimization of a smooth nonquadratic functional J over a convex and closed subset of a Banach space V. To this end we firstly shortly recall some basic definitions concerning differentiability of J, because these sometimes differ in the literature.

On the Methods for Optimal Shape Design

1990

A short survey of the numerical methods for solving optimal shape design problems is given.

ENTROPY SOLUTIONS IN THE STUDY OF ANTIPLANE SHEAR DEFORMATIONS FOR ELASTIC SOLIDS

2000

The concept of entropy solution was recently introduced in the study of Dirichlet problems for elliptic equations and extended for parabolic equations with nonlinear boundary conditions. The aim of this paper is to use the method of entropy solutions in the study of a new problem which arise in the theory of elasticity. More precisely, we consider here the infinitesimal antiplane shear deformation of a cylindrical elastic body subjected to given forces and in a frictional contact with a rigid foundation. The elastic constitutive law is physically nonlinear and the friction is described by a static law. We present a variational formulation of the model and prove the existence and the uniquen…