Generalized Penalty Method for Elliptic Variational–Hemivariational Inequalities

  • Yi-bin XiaoEmail author
  • Mircea Sofonea


We consider an elliptic variational–hemivariational inequality with constraints in a reflexive Banach space, denoted \(\mathcal{P}\), to which we associate a sequence of inequalities \(\{\mathcal{P}_n\}\). For each \(n\in \mathbb {N}\), \(\mathcal{P}_n\) is a variational–hemivariational inequality without constraints, governed by a penalty parameter \(\lambda _n\) and an operator \(P_n\). Such inequalities are more general than the penalty inequalities usually considered in literature which are constructed by using a fixed penalty operator associated to the set of constraints of \(\mathcal{P}\). We provide the unique solvability of inequality \(\mathcal{P}_n\). Then, under appropriate conditions on operators \(P_n\), we state and prove the convergence of the solution of \(\mathcal{P}_n\) to the solution of \(\mathcal{P}\). This convergence result extends the results previously obtained in the literature. Its generality allows us to apply it in various situations which we present as examples and particular cases. Finally, we consider a variational–hemivariational inequality with unilateral constraints which arises in Contact Mechanics. We illustrate the applicability of our abstract convergence result in the study of this inequality and provide the corresponding mechanical interpretations.


Variational–hemivariational inequality Clarke subdifferential Penalty method Convergence Frictional contact 

Mathematics Subject Classification

49J40 47J20 74M10 74M15 



This research was supported by the National Natural Science Foundation of China (11771067), the Applied Basic Project of Sichuan Province (2019YJ0204) and the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No 823731 CONMECH.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Electronic Science and Technology of ChinaChengduPeople’s Republic of China
  2. 2.Laboratoire de Mathématiques et PhysiqueUniversity of PerpignanPerpignanFrance

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