Convergence Results for Elliptic Variational-Hemivariational Inequalities

2021

J Optim Theory Appl

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We consider an abstract minimization problem in reflexive Banach spaces together with a specific family of approximating sets, constructed by perturbing the cost functional and the set of constraints. For this problem, we state and prove various well-posedness results in the sense of Tykhonov, under different assumptions on the data. The proofs are based on arguments of lower semicontinuity, compactness and Mosco convergence of sets. Our results are useful in the study of various mathematical models in contact mechanics. To provide examples, we introduce 2 models, which describe the equilibrium of an elastic body in contact with a rigid body covered by a rigid-plastic and an elastic material, respectively. The weak formulation of each model is in the form of a minimization problem for the displacement field. We use our abstract well-posedness results in the study of these minimization problems. In this way, we obtain existence, uniqueness and convergence results, and moreover, we provide their mechanical interpretations. KeywordsMinimization problem • Tykhonov well-posedness • Contact problem • Elastic material • Convergence results Mathematics Subject Classification 49J45 • 90C48 • 90J31 Communicated by Paolo Vannucci.

Section: Existence Uniqueness and Convergence Resultsmentioning

confidence: 99%

“…We now turn to the contact condition (31), which is described by the maximal monotone multivalued relation between the normal displacement and the opposite of the normal stress represented in Fig. 2 and which was used in a large number of papers, including [8,36]. This condition can be derived in the following way.…”

Section: The Contact Modelsmentioning

confidence: 99%

Well-Posedness of Minimization Problems in Contact Mechanics

2021

J Optim Theory Appl

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“…First, the convergence of the solution of inequality (38) to the solution of inequality (2) was obtained in [30,32] under different assumptions on the data. The aim of these papers was to obtain results on the variational-hemivariational inequality (2). In contrast, in the current paper, we focus on the hemivariational inequality (1), and we consider its perturbation (38) in order to show that its solution approaches the solution of (1) as n → ∞.…”

Section: The Case Of Hemivariational Inequalitiesmentioning

confidence: 99%

“…It deals with the analysis of various models of contact, which are expressed in terms of strongly elliptic, time-dependent or evolutionary nonlinear boundary value problems. References in the field include [4,5,10,14,20,22] and, more recently, [2,3,17,[23][24][25]. There various existence and uniqueness results have been proved by using arguments of variational and hemivariational inequalities.…”

Section: Introductionmentioning

confidence: 99%

Tykhonov triples and convergence results for hemivariational inequalities

2021

NAMC

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Consider an abstract Problem P in a metric space (X; d) assumed to have a unique solution u. The aim of this paper is to compare two convergence results u'n → u and u''n → u, both in X, and to construct a relevant example of convergence result un → u such that the two convergences above represent particular cases of this third convergence. To this end, we use the concept of Tykhonov triple. We illustrate the use of this new and nonstandard mathematical tool in the particular case of hemivariational inequalities in reflexive Banach space. This allows us to obtain and to compare various convergence results for such inequalities. We also specify these convergences in the study of a mathematical model, which describes the contact of an elastic body with a foundation and provide the corresponding mechanical interpretations.

“…They model the equilibrium of elastic bodies acted upon the body forces and surface tractions, in frictional or frictionless contact with an obstacle. References in the field are [15,16] and, more recently [2,13,18,22,28,29]. There, existence and uniqueness results for inequality problems of the form (1.1) can be found, under various assumptions on the data.…”

Section: Introductionmentioning

confidence: 99%

A Tykhonov-type well-posedness concept for elliptic hemivariational inequalities

2020

Z. Angew. Math. Phys.

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In this paper, we introduce a new Tykhonov-type well-posedness concept for elliptic hemivariational inequalities, governed by an approximating function h. We characterize the well-posedness in terms of the metric properties of the family of approximating sets, under various assumptions on h. Then, we use the well-posedness properties in order to obtain convergence results of the solution with respect to the data. The proofs are based on arguments of monotonicity combined with the properties of the Clarke directional derivative. Our results provide mathematical tools in the study of a large number of static problems in Contact Mechanics. To provide an example, we consider a mathematical model which describes the equilibrium of a rod-spring system with unilateral constraints. We prove the unique weak solvability of the model, and then we illustrate our abstract convergence results in the study of this contact problem and provide the corresponding mechanical interpretations.