# Publications of Nicusor Costea

Antiplane shear deformation of piezoelectric bodies in contact with a conductive support

We consider a mathematical model which describes the frictional contact between a piezoelectric body and an electrically conductive support. We model the material’s behavior with an electro-elastic constitutive law; the frictional contact is described with a boundary condition involving Clarke’s generalized gradient and the electrical condition on the contact surface is modeled using the subdifferential of a proper, convex and lower semicontinuous function.We derive a variational formulation of the model and then, using a fixed point theorem for set valued mappings, we prove the existence of at least one weak solution. Finally, the uniqueness of the solution is discussed; the investigation is based on arguments in the theory of variational-hemivariational inequalities.

Systems of nonlinear hemivariational inequalities and applications

In this paper we prove several existence results for a general class of systems of nonlinear hemivariational inequalities by using a fixed point theorem of Lin [21]. Our analysis includes both the cases of bounded and unbounded closed convex subsets in real reflexive Banach spaces. In the last section we apply the abstract results obtained to extend some results concerning nonlinear hemivariational inequalities, to establish existence results of Nash generalized derivative points and to prove the existence of at least one weak solution for an electroelastic contact problem.

A multiplicity result for an elliptic anisotropic differential inclusion involving variable exponents

In this paper we are concerned with the study of a class of quasilinear elliptic differential inclusions involving the anisotropic $\overrightarrow{p}(\cdot)$-Laplace operator, on a bounded open subset of $IR^n$ which has a smooth boundary. The abstract framework required to study this kind of dierential inclusions lies at the interface of three important branches in analysis: nonsmooth analysis, the variable exponent Lebesgue-Sobolev spaces theory and the anisotropic Sobolev spaces theory. Using the concept of nonsmooth critical point we are able to prove that our problem admits at least two non-trivial weak solutions.

Contact models leading to variational–hemivariational inequalities

A frictional contact model, under the small deformations hypothesis, for static processes is considered. We model the behavior of the material by a constitutive law using the subdifferential of a proper, convex and lower semicontinuous function. The contact is described with a boundary condition involving Clarke’s generalized gradient. Our study focuses on the weak solvability of the model. Based on a fixed point theorem for setvalued mappings, we prove the existence of at least one weak solution. The uniqueness, the boundedness and the stability of the weak solution are also discussed; the investigation is based on arguments in the theory of variational–hemivariational inequalities. Finally, we present several examples of constitutive laws and friction laws for which our theoretical results are valid.

Inequality problems of quasi-hemivariational type involving set-valued operators and a nonlinear term

The aim of this paper is to establish the existence of at least one solution for a general inequality of quasi-hemivariational type, whose solution is sought in a subset K of a real Banach space E. First, we prove the existence of solutions in the case of compact convex subsets and the case of bounded closed and convex subsets. Finally, the case when K is the whole space is analyzed and necessary and sufficient conditions for the existence of solutions are stated. Our proofs rely essentially on the Schauder’s fixed point theorem and a version of the KKM principle due to Ky Fan (Math Ann 266:519–537, 1984).

Elliptic boundary value problems with nonsmooth potential and mixed boundary conditions

The goal of this article is to establish the existence of at least one solution for a boundary value problem governed by a quasilinear elliptic operator and two multivalued functions given by Clarke’s generalized gradient of some locally Lipschitz functionals. We divide the boundary $\Gamma_1$ of our domain into two open measurable parts $\Gamma_1$ and $\Gamma_2$ and we impose a nonhomogeneous Neumann boundary condition on $\Gamma_1$, while on $\Gamma_2$ we impose a Dirichlet boundary condition. This kind of problems have been treated in the past by various authors. However, in all the work we are aware of either a Neumann or a Dirichlet boundary condition imposed on the entire boundary. Another main point of interest of this article is that our problem depends on a real parameter $\lambda>0$ and we are able to prove the existence of solutions for any $\lambda>0$.

Variational-like inequality problems involving set-valued maps and generalized monotonicity

The aim of this paper is to establish existence results for some variational-like inequality problems involving set-valued maps. We study both the case of reflexive and nonreflexive Banach spaces. When the set $K$ in which we seek solutions is compact and convex we no dot impose any monotonicity assumptions on the set-valued map $A$ which appears in the formulation of the inequality problems. In the case when $K$ is only bounded closed and convex certain monotonicity assumptions are needed: we ask the set-valued map $A$ to be relaxed $\eta-\alpha$ monotone in order to prove the existence of at least one solution for generalized variational-like inequalities (see problem (2) in Section 1) and relaxed $\eta-\alpha$ quasimonotone in order to establish existence results for variational-like inequalities (see problem (3) in Section 1). We also provide sufficient conditions for the existence of solutions in the case when $K$ is unbounded closed and convex. Our results generalize and improve some known results in the literature on this topic.

Existence and uniqueness results for a class of quasi-hemivariational inequalities

In this paper we are concerned with the study of a nonstandard quasi-hemivariational inequality. Using a fixed point theorem for set-valued mappings the existence of at least one solution in bounded closed and convex subsets is established. We also provide sufficient conditions for which our inequality possesses solutions in the case of unbounded sets. Finally, the uniqueness and the stability of the solution are analyzed in a particular case.

Nonlinear hemivariational inequalities and applications to nonsmooth mechanics

The goal of this paper is to establish several existence results for a class of nonstandard hemivariational inequalities. Our analysis includes both the cases of bounded and unbounded closed and convex subsets in real reflexive Banach spaces. The proofs strongly rely on the KKM Principle combined with the Mosco Alternative. In the last section of the paper several applications illustrate the abstract results that were proved throughout the paper.

Weak solutions for nonlinear antiplane problems leading to hemivariational inequalities

We analyze the antiplane shear deformation of an elastic cylinder in frictional contact with a rigid foundation, for static processes, under the small deformation hypothesis. Based on the Knaster-Kuratowski-Mazurkiewicz technique in the theory of the hemivariational inequalities, we prove that the model has at least one weak solution. Moreover, we present several examples of constitutive laws and friction laws for which our theoretical results are valid. Finally, we comment on the conditions which guarantee the uniqueness of the solution.

Hartman-Stampacchia results for stably pseudomonotone operators and non-linear hemivariational inequalities

We are concerned with two classes of non-standard hemivariational inequalities. In the first case we establish a Hartman–Stampacchia type existence result in the framework of stably pseudomonotone operators. Next, we prove an existence result for a class of non-linear perturbations of canonical hemivariational inequalities. Our analysis includes both the cases of compact sets nd of closed convex sets in Banach spaces. Applications to non-coercive hemivariational and variational-hemivariational inequalities illustrate the abstract results of this article.

On a class of variational-hemivariational inequalities involving set valued mappings

Using the KKM technique, we establish some existence results for variational-hemivariational inequalities involving monotone set valued mappings on bounded, closed and convex subsets in reflexive Banach spaces. We also derive several sufficient conditions for the existence of solutions in the case of unbounded subsets.

Existence results for hemivariational inequalities involving relaxed $\eta - \alpha$ monotone mappings

We establish some existence results for hemivariational inequalities with relaxed $\eta-\alpha$ monotone mappings on bounded, closed and convex subsets in reflexive Banach spaces. Our proofs rely essentially on a fixed point theorem for set valued mappings which is due to Tarafdar. We also give a sufficient condition for the existence of solutions in the case of unbounded subsets.

Nonlinear, degenerate and singular eigenvalue problems on $IR^N$

The goal of this paper is to establish the existence of a continuous family of eigenvalues lying in a neighborhood at the right of the origin for some nonlinear, nonhomogeneous, degenerate and singular elliptic operators on the whole space $IR^N$ . Our proofs rely essentially on the CaffarelliKohnNirenberg inequality combined with the Banach fixed point theorem.

Continuous family of eigenvalues concentrating in a small neighborhood at the right of the origin for a class of discrete boundary value problems

In this paper, we prove the existence of a continuous spectrum that lies in a neighborhood at the right of the origin for some nonlinear difference operators. Our proofs rely essentially on the Banach fixed point theorem and a minimization technique.