Elliptic Hemivariational Inequalities Research Articles

Superlinear elliptic hemivariational inequalities

We study a nonlinear nonhomogeneous Dirichlet problem with a nonsmooth potential which is superlinear but without satisfying the Ambrosetti-Rabinowitz condition. Using the nonsmooth critical point theory and critical groups we prove two multiplicity theorems producing three and five solutions respectively. In the second multiplicity theorem, we provide sign information for all the solutions and the solutions are ordered.

The interior penalty virtual element method for the fourth-order elliptic hemivariational inequality

We develop an interior penalty virtual element method (IPVEM) for solving a Kirchhoff plate contact problem, which can be described by a fourth-order elliptic hemivariational inequality (HVI). The virtual element space in IPVEM is constructed by modifying the H2-conforming virtual element space, and the number of degrees of freedom is greatly reduced. To force the C1 continuity, the interior penalty scheme is adopted, that is, we introduce a penalty term for the jump of normal derivatives on mesh edges and two additional terms to guarantee the consistency and symmetry of the scheme. With certain assumptions, the well-posedness of the discrete problem is proved. Furthermore, a priori error estimation is established for the IPVEM for the fourth-order elliptic HVI, and we show that the lowest-order VEM achieves optimal convergence order. Finally, some numerical examples are presented to support the theoretical results.

On variational–hemivariational inequalities in Banach spaces

This paper is devoted to a well-posedness analysis of elliptic variational–hemivariational inequalities in Banach spaces. The differential operator associated with the variational–hemivariational inequality is assumed to be strongly monotone of a general order, in contrast to that in the majority of existing references on this subject where the differential operator is assumed to be strongly monotone of order 2. Moreover, the solution existence is proved with an approach more accessible to applied mathematicians and engineers, instead of through an abstract surjectivity result for pseudomonotone operators in existing references. Equivalent minimization principles are established for certain variational–hemivariational inequalities, which are valuable for developing efficient numerical algorithms. The theoretical results are applied to the analysis of a mixed hemivariational inequality in the study of a generalized Newtonian fluid flow problem involving a nonsmooth slip boundary condition of friction type. Existence and uniqueness of both the velocity and pressure unknowns are shown for the mixed hemivariational inequalities.

The virtual element method for general variational–hemivariational inequalities with applications to contact mechanics

This paper mainly analyzes the general elliptic variational–hemivariational inequalities with or without constraints by using the virtual element method. The approximations can be internal or external and a Céa’s type inequality is derived for a priori error estimates. Then, we apply the results to a variational–hemivariational inequality arising in frictional contact problems, and the optimal order error estimate is obtained for the linear virtual element solution under appropriate solution regularity assumptions. Finally, numerical simulation results are reported to show the performance of the proposed method, in particular, numerical convergence orders are in good agreement with the theoretical predictions.

A new class of history-dependent quasi variational–hemivariational inequalities with constraints

In this paper we consider an abstract class of time-dependent quasi variational–hemivariational inequalities which involves history-dependent operators and a set of unilateral constraints. First, we establish the existence and uniqueness of solution by using a recent result for elliptic variational–hemivariational inequalities in reflexive Banach spaces combined with a fixed-point principle for history-dependent operators. Then, we apply the abstract result to show the unique weak solvability to a quasistatic viscoelastic frictional contact problem. The contact law involves a unilateral Signorini-type condition for the normal velocity and the nonmonotone normal damped response condition while the friction condition is a version of the Coulomb law of dry friction in which the friction bound depends on the accumulated slip.

A Deep Learning Method for Elliptic Hemivariational Inequalities

A Deep Learning Method for Elliptic Hemivariational Inequalities

Hemivariational inequalities on graphs

In this paper, a new class of hemivariational inequalities is introduced. It concerns Laplace operator on locally finite graphs together with multivalued nonmonotone nonlinearities expressed in terms of Clarke’s subdifferential. First of all, we state and prove some results on the subdifferentiability of nonconvex functionals defined on graphs. Thereafter, an elliptic hemivariational inequality on locally finite graphs is considered and the existence and uniqueness of its weak solutions are proved by means of the well-known surjectivity result for pseudomonotone mappings. In the end of this paper, we tackle the problem of hemivariational inequalities of parabolic type on locally finite graphs and we prove the existence of its weak solutions.

A Class of Variational\u2013Hemivariational Inequalities for Bingham Type Fluids

In this paper we investigate a new class of elliptic variational–hemivariational inequalities without the relaxed monotonicity condition of the generalized subgradient. The inequality describes the mathematical model of the steady state flow of incompressible fluid of Bingham type in a bounded domain. The boundary condition represents a generalization of the no leak condition, and a multivalued and nonmonotone version of a nonlinear Navier–Fujita frictional slip condition. The analysis provides results on existence of solution to a variational–hemivariational inequality, continuous dependence of the solution on the data, existence of solutions to optimal control problems, and the dependence of the solution on the yield limit. The proofs profit from results of nonsmooth analysis and the theory of multivalued pseudomontone operators.

Numerical solution of anH(curl)-elliptic hemivariational inequality

AbstractThis paper is concerned with the analysis and numerical solution of an $\boldsymbol {H}({\textbf {curl}})$-elliptic hemivariational inequality (HVI). One source of the HVI is through a temporal semidiscretization of a related hyperbolic Maxwell equation problem. An equivalent minimization principle is introduced, and the solution existence and uniqueness of the $\boldsymbol {H}({\textbf {curl}})$-elliptic HVI are proved. Numerical analysis of the HVI is provided with a general Galerkin approximation, including a Céa’s inequality for convergence and error estimation. When the linear edge finite element method is employed, an optimal-order error estimate is derived under a suitable solution regularity assumption. A fully discrete scheme based on the backward Euler difference in time and a mixed finite element method in space is also analyzed, and stability estimates are derived for first-order terms of the fully discrete solution. Numerical results are reported on linear edge finite element solutions of the $\boldsymbol {H}({\textbf {curl}})$-elliptic HVI for numerical evidence of the theoretically predicted convergence order.

Well-posedness of a general class of elliptic mixed hemivariational–variational inequalities

In this paper, well-posedness of a general class of elliptic mixed hemivariational–variational inequalities is studied. This general class includes several classes of the previously studied elliptic mixed hemivariational–variational inequalities as special cases. Moreover, our approach of the well-posedness analysis is easily accessible, unlike those in the published papers on elliptic mixed hemivariational–variational inequalities so far. First, prior theoretical results are recalled for a class of elliptic mixed hemivariational–variational inequalities featured by the presence of a potential operator. Then the well-posedness results are extended through a Banach fixed-point argument to the same class of inequalities without the potential operator assumption. The well-posedness results are further extended to a more general class of elliptic mixed hemivariational–variational inequalities through another application of the Banach fixed-point argument. The theoretical results are illustrated in the study of a contact problem. For comparison, the contact problem is studied both as an elliptic mixed hemivariational–variational inequality and as an elliptic variational–hemivariational inequality.

Numerical analysis of a general elliptic variational-hemivariational inequality

Numerical analysis of a general elliptic variational-hemivariational inequality

The Interior Penalty Virtual Element Method for the Fourth-Order Elliptic Hemivariational Inequality

The Interior Penalty Virtual Element Method for the Fourth-Order Elliptic Hemivariational Inequality

Simultaneous distributed and Neumann boundary optimal control problems for elliptic hemivariational inequalities

Simultaneous distributed and Neumann boundary optimal control problems for elliptic hemivariational inequalities

Minimization arguments in analysis of variational–hemivariational inequalities

In this paper, an alternative approach is provided in the well-posedness analysis of elliptic variational–hemivariational inequalities in real Hilbert spaces. This includes the unique solvability and continuous dependence of the solution on the data. In most of the existing literature on elliptic variational–hemivariational inequalities, well-posedness results are obtained by using arguments of surjectivity for pseudomonotone multivalued operators, combined with additional compactness and pseudomonotonicity properties. In contrast, following (Han in Nonlinear Anal B Real World Appl 54:103114, 2020; Han in Numer Funct Anal Optim 42:371–395, 2021), the approach adopted in this paper is based on the fixed point structure of the problems, combined with minimization principles for elliptic variational–hemivariational inequalities. Consequently, only elementary results of functional analysis are needed in the approach, which makes the theory of elliptic variational–hemivariational inequalities more accessible to applied mathematicians and engineers. The theoretical results are illustrated on a representative example from contact mechanics.

An inverse problem for Bingham type fluids

In this paper we study a coefficient identification problem described by an elliptic variational-hemivariational inequality with unilateral constraints. The inequality is the weak formulation of the mathematical model of a stationary incompressible flow of Bingham type in a bounded domain. The unknown coefficient is a generalized viscosity function of the fluid. The boundary conditions represent generalizations of the no leak condition and a multivalued and nonmonotone version of a nonlinear Navier–Fujita frictional slip condition. The result on well posedness of the direct problem is established based on the theory of multivalued pseudomonotone operators. The existence to the inverse problem is proved by a Weierstrass type argument.

Distributed optimal control problems for a class of elliptic hemivariational inequalities with a parameter and its asymptotic behavior

In this paper, we study optimal control problems on the internal energy for a system governed by a class of elliptic boundary hemivariational inequalities with a parameter. The system has been originated by a steady-state heat conduction problem with non-monotone multivalued subdifferential boundary condition on a portion of the boundary of the domain described by the Clarke generalized gradient of a locally Lipschitz function. We prove an existence result for the optimal controls and we show an asymptotic result for the optimal controls and the system states, when the parameter, like a heat transfer coefficient, tends to infinity on a portion of the boundary.

Existence, Comparison, and Convergence Results for a Class of Elliptic Hemivariational Inequalities

In this paper we study a class of elliptic boundary hemivariational inequalities which originates in the steady-state heat conduction problem with nonmonotone multivalued subdifferential boundary condition on a portion of the boundary described by the Clarke generalized gradient of a locally Lipschitz function. First, we prove a new existence result for the inequality employing the theory of pseudomonotone operators. Next, we give a result on comparison of solutions, and provide sufficient conditions that guarantee the asymptotic behavior of solution, when the heat transfer coefficient tends to infinity. Further, we show a result on the continuous dependence of solution on the internal energy and heat flux. Finally, some examples of convex and nonconvex potentials illustrate our hypotheses.

The Virtual Element Method for an Elliptic Hemivariational Inequality with Convex Constraint

The Virtual Element Method for an Elliptic Hemivariational Inequality with Convex Constraint

A Revisit of Elliptic Variational-Hemivariational Inequalities

In this paper, we provide an alternative approach to establish the solution existence and uniqueness for elliptic variational-hemivariational inequalities. The new approach is based on elementary results from functional analysis, and thus removes the need of the notion of pseudomonotonicity and the dependence on surjectivity results for pseudomonotone operators. This makes the theory of elliptic variational-hemivariational inequalities more accessible to applied mathematicians and engineers. In addition, equivalent minimization principles are further explored for particular elliptic variational-hemivariational inequalities. Representative examples from contact mechanics are discussed to illustrate application of the theoretical results.

The virtual element method for general elliptic hemivariational inequalities

An abstract framework of the virtual element method is established for solving general elliptic hemivariational inequalities with or without constraint, and a unified a priori error analysis is given for both cases. Then, virtual element methods of arbitrary order are applied to solve three elliptic hemivariational inequalities arising in contact mechanics, and optimal order error estimates are shown with the linear virtual element solutions. Numerical simulation results are reported in several contact problems; in particular, the numerical convergence orders of the lowest order virtual element solutions are shown to be in good agreement with the theoretical predictions.