Second-order asymptotic analysis for noncoercive convex optimization

We use second-order asymptotic analysis to deal with the minimization problem of a noncoercive convex function in a reflexive Banach space. To that end, we first introduce the definition of a second-order asymptotic cone, and its respective function, based on previous results for the finite dimensional case. We provide necessary and sufficient conditions for the existence of solutions for noncoercive convex minimization problems. Examples for which our assumptions are easier to verify than other well-known results are also provided.     

Equilibrium problems and noncoercive variational inequalities

In this paper, we are interested to establish existence results for equilibrium problems in a noncoercive framework by using techniques of recession analysis. The abstract result is then applied to find solution of noncoercive monotone and pseudomonotone variational inequalities     

One-sided contact problems with friction arising along the normal

We study a boundary contact problem for a micropolar homogeneous elastic hemitropic medium with regard of friction; in the considered case, friction forces do not arise in the tangential displacement but correspond to a normal displacement of the medium. We consider two cases: the coercive case (in which the elastic body has a fixed part of the boundary) and the noncoercive case (without fixed parts). By using the Steklov–Poincaré operator, we reduce this problem to an equivalent boundary variational inequality. Existence and uniqueness theorems are proved for the weak solution on the basis of properties of general variational inequalities. In the coercive case, the problem is unconditionally solvable, and the solution depends continuously on the data of the original problem. In the noncoercive case, we present closed-form necessary conditions for the existence of a solution of the contact problem. Under additional assumptions, these conditions are also sufficient for the existence of a solution.     

On the Minimal Solutions of Variational Inequalities in Orlicz-Sobolev Spaces

In this paper, the author studies the existence of the minimal nonnegative solutions of some elliptic variational inequalities in Orlicz-Sobolev spaces on bounded or unbounded domains. She gets some comparison results between different solutions as tools to pass to the limit in the problems and to show the existence of the minimal solutions of the variational inequalities on bounded domains or unbounded domains. In both cases,coercive and noncoercive operators are handled. The sufficient and necessary conditions for the existence of the minimal nonnegative solution of the noncoercive variational inequality on bounded domains are established.     

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.     

Noncoercive mixed equilibrium problems under pseudomonotone perturbations and applications to nonlinear evolution equations with lack of coercivity

In this paper, we study the existence of solutions for noncoercive mixed equilibrium problems which are described by the sum of a maximal monotone bifunction and a pseudomonotone (or quasimonotone) bifunction in the sense of Brézis. Our approach is based on recession analysis and on recent results established by the authors for the existence of solutions of mixed equilibrium problems under pseudomonotone perturbations. As an application, we study the existence of solutions for nonlinear evolution equations associated with a noncoercive time-dependent pseudomonotone (or quasimonotone) operator.     

Elliptic variational hemivariational inequalities

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

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.     

Existence and comparison results for noncoercive and nonmonotone multivalued elliptic problems

We consider noncoercive quasilinear elliptic inclusions under multivalued flux boundary conditions involving multifunctions of Clarke’s generalized gradient. Our main goal is to provide existence and comparison results which are based on an appropriate generalization of the notions of subsolutions and supersolutions. Furthermore, compactness and extremality results of the solution set enclosed by an ordered pair of sub–supersolutions will be proved.     

On Some Noncoercive Variational Inequalities

We study existence and regularity of solutions of noncoercive variational inequalities.     

一类非强制的变分不等式

讨论了一类非强制的变分不等式解之存在性的一个必要条件和一个充分条件,改进了文[1]中的相应结果。     

Existence of Solutions for Noncoercive Hemivariational Inequalities by an Equilibrium Approach Under Pseudomonotone Perturbation

In this paper, we study the existence of a solution for a hemivariational inequality problem in a noncoercive framework. The approach adopted is an equilibrium problem formulation associated with a maximal monotone bifunction with pseudomonotone perturbation. We proceed by introducing auxiliary problems that will be studied using a new existence result for equilibrium problems. An example to illustrate the use of the theory is given.     

Constant Sign and Sign Changing Solutions for Systems of Quasilinear Elliptic Equations

We establish the existence of two opposite constant sign solutions for a general noncoercive quasilinear elliptic system with homogeneous Dirichlet boundary conditions. In the case where the system has a variational structure, by strengthening the hypotheses, we obtain a third nontrivial solution which is sign changing in the sense that one cannot have both components of the new solution of the same constant sign. Our approach relies on a suitable method of sub-supersolutions combined with truncation and variational arguments that does not require a subcritical growth condition.     

A nontrivial solution of mountain-pass type for a hemivariational inequality

We consider a class of noncoercive hemivariational inequalities involving the p-Laplacian. Our goal is to obtain the existence of a nontrivial solution. Using the mountain-pass theorem for locally Lipschitz functionals we obtain the desired result.     

Existence of positive solutions for a class of indefinite elliptic problems in {\mathbb R}^N

We consider the question of existence of positive solutions for a class of elliptic problems in all of having a noncoercive linear part and a sign-changing nonlinearity of the form . Received December 20, 1999 / Accepted July 17, 2000 /Published online November 9, 2000     

On Neumann problems for nonlocal Hamilton–Jacobi equations with dominating gradient terms

We are concerned with the well-posedness of Neumann boundary value problems for nonlocal Hamilton–Jacobi equations related to jump processes in general smooth domains. We consider a nonlocal diffusive term of censored type of order strictly less than 1 and Hamiltonians both in coercive form and in noncoercive Bellman form, whose growth in the gradient make them the leading term in the equation. We prove a comparison principle for bounded sub-and supersolutions in the context of viscosity solutions with generalized boundary conditions, and consequently by Perron’s method we get the existence and uniqueness of continuous solutions. We give some applications in the evolutive setting, proving the large time behaviour of the associated evolutive problem under suitable assumptions on the data.     

An extension theorem in SBV and an application to the homogenization of the Mumford–Shah functional in perforated domains

The aim of this paper is to prove the existence of extension operators for SBV functions from periodically perforated domains. This result will be the fundamental tool to prove the compactness in a noncoercive homogenization problem.     

On the solvability of noncoercive linear variational inequalities in separable Hilbert spaces

In this paper, we build an existence theory for linear variational inequalities associated with an operator which generalizes in Hilbert space the class of copositive plus matrices. We show how this theory can be used to study some important engineering problems governed by noncoercive variational inequalities.Thanks are due to Professor V. H. Nguyen for many valuable discussions. The author thanks the Associate Editor and the referees for their helpful suggestions     

Noncoercive Problems in Nonlinear Elasticity with Stored Energies Having Fast or Slow Growth: Orlicz—Sobolev Space Setting

This paper is concerned with a noncoercive unilateral problem in nonlinear elasticity with stored energy function having fast or slow growth. We prove an existence result for the associated minimization problem by using a recession approach in Orlicz—Sobolev space.     

Augmented Lagrangian functions for constrained optimization problems

In this paper, in order to obtain some existence results about solutions of the augmented Lagrangian problem for a constrained problem in which the objective function and constraint functions are noncoercive, we construct a new augmented Lagrangian function by using an auxiliary function. We establish a zero duality gap result and a sufficient condition of an exact penalization representation for the constrained problem without the coercive or level-bounded assumption on the objective function and constraint functions. By assuming that the sequence of multipliers is bounded, we obtain the existence of a global minimum and an asymptotically minimizing sequence for the constrained optimization problem.