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Well-Posedness of Minimization Problems in Contact Mechanics

Author

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  • Mircea Sofonea

    (University of Electronic Science and Technology of China
    University of Perpignan Via Domitia)

  • Yi-bin Xiao

    (University of Electronic Science and Technology of China)

Abstract

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.

Suggested Citation

  • Mircea Sofonea & Yi-bin Xiao, 2021. "Well-Posedness of Minimization Problems in Contact Mechanics," Journal of Optimization Theory and Applications, Springer, vol. 188(3), pages 650-672, March.
    • Handle: RePEc:spr:joptap:v:188:y:2021:i:3:d:10.1007_s10957-020-01801-y
    DOI: 10.1007/s10957-020-01801-y
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    References listed on IDEAS

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    1. Fang, Ya-Ping & Huang, Nan-Jing & Yao, Jen-Chih, 2010. "Well-posedness by perturbations of mixed variational inequalities in Banach spaces," European Journal of Operational Research, Elsevier, vol. 201(3), pages 682-692, March.
    2. Mircea Sofonea & Yi-bin Xiao, 2019. "On the Well-Posedness Concept in the Sense of Tykhonov," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 139-157, October.
    3. X. X. Huang, 2001. "Extended and strongly extended well-posedness of set-valued optimization problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 53(1), pages 101-116, April.
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