A GENERALIZED PENALTY METHOD FOR DIFFERENTIAL VARIATIONAL-HEMIVARIATIONAL INEQUALITIES

doi:10.1007/s10473-022-0114-z

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (1): 247-264.doi: 10.1007/s10473-022-0114-z

A GENERALIZED PENALTY METHOD FOR DIFFERENTIAL VARIATIONAL-HEMIVARIATIONAL INEQUALITIES

卢亮¹, 李丽洁², Mircea SOFONEA³

1. Guangxi Key Laboratory of Cross-border E-commerce Intelligent Information Processing, Guangxi University of Finance and Economics, Nanning, 530003, China;
2. Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin, 537000, China;
3. Laboratoire de Mathématiques et Physique, Université de Perpignan Via Domitia, 52 Avenue Paul Alduy 66, 860, Perpignan, France

收稿日期:2020-07-19 修回日期:2021-05-26 出版日期:2022-02-25 发布日期:2022-02-24
通讯作者: Mircea SOFONEA,E-mail:sofonea@univ-perp.fr E-mail:sofonea@univ-perp.fr
作者简介:Liang LU,E-mail:gxluliang@163.com;Lijie LI,E-mail:lilijiemaths@126.com
基金资助:
This work was supported by the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement (823731 CONMECH). The first author is also supported by National Natural Science Foundation of China (11671101), Guangxi Natural Science Foundation (2021GXNSFAA075022) and Project of Guangxi Education Department (2020KY16017). The second author is also supported by National Natural Science Foundation of China (11961074) and Yulin normal university of scientific research fund for high-level talents (G2019ZK39, G2021ZK06).

A GENERALIZED PENALTY METHOD FOR DIFFERENTIAL VARIATIONAL-HEMIVARIATIONAL INEQUALITIES

Liang LU¹, Lijie LI², Mircea SOFONEA³

1. Guangxi Key Laboratory of Cross-border E-commerce Intelligent Information Processing, Guangxi University of Finance and Economics, Nanning, 530003, China;
2. Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin, 537000, China;
3. Laboratoire de Mathématiques et Physique, Université de Perpignan Via Domitia, 52 Avenue Paul Alduy 66, 860, Perpignan, France

Received:2020-07-19 Revised:2021-05-26 Online:2022-02-25 Published:2022-02-24
Contact: Mircea SOFONEA,E-mail:sofonea@univ-perp.fr E-mail:sofonea@univ-perp.fr
Supported by:
This work was supported by the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement (823731 CONMECH). The first author is also supported by National Natural Science Foundation of China (11671101), Guangxi Natural Science Foundation (2021GXNSFAA075022) and Project of Guangxi Education Department (2020KY16017). The second author is also supported by National Natural Science Foundation of China (11961074) and Yulin normal university of scientific research fund for high-level talents (G2019ZK39, G2021ZK06).

摘要/Abstract

摘要： We consider a differential variational-hemivariational inequality with constraints, in the framework of reflexive Banach spaces. The existence of a unique mild solution of the inequality, together with its stability, was proved in[1]. Here, we complete these results with existence, uniqueness and convergence results for an associated penalty-type method. To this end, we construct a sequence of perturbed differential variational-hemivariational inequalities governed by perturbed sets of constraints and penalty coefficients. We prove the unique solvability of each perturbed inequality as well as the convergence of its solution to the solution of the original inequality. Then, we consider a mathematical model which describes the equilibrium of a viscoelastic rod in unilateral contact. The weak formulation of the model is in a form of a differential variational-hemivariational inequality in which the unknowns are the displacement field and the history of the deformation. We apply our abstract penalty method in the study of this inequality and provide the corresponding mechanical interpretations.

关键词: differential variational-hemivariational inequality, generalized penalty method, Mosco convergence, viscoelastic rod, unilateral constraint

Abstract: We consider a differential variational-hemivariational inequality with constraints, in the framework of reflexive Banach spaces. The existence of a unique mild solution of the inequality, together with its stability, was proved in[1]. Here, we complete these results with existence, uniqueness and convergence results for an associated penalty-type method. To this end, we construct a sequence of perturbed differential variational-hemivariational inequalities governed by perturbed sets of constraints and penalty coefficients. We prove the unique solvability of each perturbed inequality as well as the convergence of its solution to the solution of the original inequality. Then, we consider a mathematical model which describes the equilibrium of a viscoelastic rod in unilateral contact. The weak formulation of the model is in a form of a differential variational-hemivariational inequality in which the unknowns are the displacement field and the history of the deformation. We apply our abstract penalty method in the study of this inequality and provide the corresponding mechanical interpretations.

Key words: differential variational-hemivariational inequality, generalized penalty method, Mosco convergence, viscoelastic rod, unilateral constraint

中图分类号:

34G20

卢亮, 李丽洁, Mircea SOFONEA. A GENERALIZED PENALTY METHOD FOR DIFFERENTIAL VARIATIONAL-HEMIVARIATIONAL INEQUALITIES[J]. 数学物理学报(英文版), 2022, 42(1): 247-264.

Liang LU, Lijie LI, Mircea SOFONEA. A GENERALIZED PENALTY METHOD FOR DIFFERENTIAL VARIATIONAL-HEMIVARIATIONAL INEQUALITIES[J]. Acta mathematica scientia,Series B, 2022, 42(1): 247-264.

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A GENERALIZED PENALTY METHOD FOR DIFFERENTIAL VARIATIONAL-HEMIVARIATIONAL INEQUALITIES

A GENERALIZED PENALTY METHOD FOR DIFFERENTIAL VARIATIONAL-HEMIVARIATIONAL INEQUALITIES

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