Solution of a semicoercive Signorini problem by a method of iterative proximal regularization of a modified Lagrange functional

Duality methods based on classical schemes for constructing Lagrange functional are inapplicable for solving semicoercive variational inequalities in mechanics. In this paper we approximately solve a scalar semicoercive Signorini problem, using a duality method based on the iterative proximal regularization of a modified Lagrange functional. We realize the algorithm with the help of the finite element method on a sequence of domain triangulations.     

Regularization in the Mosolov and Myasnikov problem with boundary friction

We propose an iterative algorithm for solving a semicoercive nonsmooth variational inequality. The algorithm is based on the stepwise partial smoothing of the minimized functional and an iterative proximal regularization method.We obtain a solution to the variational Mosolov and Myasnikov problem with boundary friction as a limit point of a sequence of solutions to stable auxiliary problems.     

On the convergence rate of the finite element method in a semicoercive problem with friction

We estimate the convergence rate of the finite element method in the minimization problem for a semicoercive nondifferentiable functional arising in the study of a model problem with friction.     

Noncoercive Variational–Hemivariational Inequalities: Existence,Approximation by Double Regularization,and Application to Nonmonotone Contact Problems

Journal of Optimization Theory and Applications - We study noncoercive nonlinear variational–hemivariational inequalities that encompass semicoercive nonlinear monotone variational...     

Iterative proximal regularization method for finding a saddle point in the semicoercive Signorini problem

An algorithm for seeking a saddle point for the semicoercive variational Signorini inequality is studied. The algorithm is based on an iterative proximal regularization of a modified Lagrangian functional.     

A method of successive approximations for solving the quasi-variational Signorini inequality

We solve the semicoercive quasi-variational Signorini inequality that corresponds to the contact problem with friction known in the elasticity theory by a method of successive approximations. For solving auxiliary problems with a given friction occurring on each outer step of the iterative process we use the Uzawa method based on iterative proximal regularization of a modified Lagrangian functional. We study the stabilization of the sequence of auxiliary finite-element solutions obtained on outer steps of the method of successive approximations and present results of numerical calculations.     

Iterative proximal regularization of the modified Lagrangian functional for solving the quasi-variational Signorini inequality

The iterative Uzawa method with a modified Lagrangian functional is used to numerically solve the semicoercive Signorini problem with friction (quasi-variational inequality).     

Stable smoothing method for solving a model mechanical problem with friction

A solution algorithm with a modified Lagrangian functional is studied as applied to a semicoercive model problem with friction.     

A minimax approach to semicoercive hemivariational inequalities

A new minimax approach for treating semicoercive hemivariational inequalities is presented. An elliptic Dirichlet problem at resonance and with discontinuous nonlinearties is solved under a recession condition     

Optimal design of an elastic beam with a unilateral elastic foundation: Semicoercive state problem

A design optimization problem for an elastic beam with a unilateral elastic foundation is analyzed. Euler-Bernoulli’s model for the beam and Winkler’s model for the foundation are considered. The state problem is represented by a nonlinear semicoercive problem of 4th order with mixed boundary conditions. The thickness of the beam and the stiffness of the foundation are optimized with respect to a cost functional. We establish solvability conditions for the state problem and study the existence of a solution to the optimization problem.     

Duality scheme for solving the semicoercive signorini problem with friction

The iterative Uzawa method with a modified Lagrangian functional is used to numerically solve the semicoercive Signorini problem with friction (quasi-variational inequality).     

The lagrange multiplier method for solving a semicoercive model problem with friction

In model problems with friction, unconditional minimization of a nondifferentiable functional is reduced to conditional minimization of a differentiable functional. To solve the semicoercive problem obtained, we use a dual scheme based on a modified Lagrangian functional.     

On the convergence of the Uzawa method with a modified Lagrangian functional for variational inequalities in mechanics

For the scalar semicoercive Signorini problem, the dual problem is constructed on the basis of a modified Lagrangian functional. The convergence of the gradient method as applied to solving the dual problem is examined.     

Solving the quasi-variational Signorini inequality by the method of successive approximations

The method of successive approximations is examined as a tool for solving the semicoercive quasi-variational Signorini inequality. The auxiliary problems with given friction arising at each step of this method are solved using the Uzawa method with an iterative proximal regularization of the modified Lagrangian functional.     

New affine measures of symmetry for convex bodies

Grünbaum introduced measures of symmetry for convex bodies that measure how far a given convex body is from a centrally symmetric one. Here, we introduce new measures of symmetry that measure how far a given convex body is from one with “enough symmetries”.To define these new measures of symmetry, we use affine covariant points. We give examples of convex bodies whose affine covariant points are “far apart”. In particular, we give an example of a convex body whose centroid and Santaló point are “far apart”.     

Iterative proximal regularization of a modified Lagrangian functional for solving a semicoercive model problem with friction

A problem of unconstrained minimization of a semicoercive nondifferentiable functional corresponding to a model friction problem is reduced to a problem of constrained minimization of a differentiable functional. An algorithm based on iterative proximal regularization of a modified Lagrangian functional is used for solving the problem thus obtained. Convergence of a finite-element solution is investigated. Results of numerical calculations are presented.     

Semicoercive variational hemivariational inequalities

The aim of this paper is the mathematical study of a general class of semicoercive variational hemivariational inequalities introduced by P.D. Panagiotopoulos in order to formulate problems of mechanics involving nonconvex and nonsmooth energy function. Our approach is based on the asymptotic behavior of the functions which are involved in the variational problems.     

Semicoercive Variational-hemivariational Inequalities with Unilateral Growth Conditions

A variational-hemivariational inequality on a vector valued functions space is studied with the nonlinear part satisfying the unilateral growth condition. The higher order term is assumed to be pseudo-monotone and semicoercive. The existence result has been obtained by making use of the theory of pseudo-monotone operators combined with the Lipschitz regulariza-tion of the nonlinear part.     

Fixed points of non-Hamiltonian symplectomorphisms

In this article we study a version of the Arnold conjecture for symplectic maps that are not Hamiltonian. That is, we give a lower bound for the number of fixed points such a map must have. We achieve the result for symplectic maps with sufficiently small Calabi invariant.     

Jackson’s integral of the Hurwitz zeta function

We give a Jacksonq-integral analogue of Euler’s logarithmic sine integral established in 1769 from several points of view, specifically from the one relating to the Hurwitz zeta function. Partially supported by Grant-in-Aid for Exploratory Research No. 15654003. Partially supported by Grant-in-Aid for Scientific Research (B) No. 15340003. Partially supported by Grant-in-Aid for Scientific Research (B) No. 15340012.