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Well-Posedness of Minimization Problems in Contact Mechanics

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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.

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Acknowledgements

This research was supported by the National Natural Science Foundation of China (11771067), the Applied Basic Project of Sichuan Province (2019YJ0204), the Fundamental Research Funds for the Central Universities (ZYGX2019J095) and the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No 823731 CONMECH.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, People’s Republic of China

    Mircea Sofonea & Yi-bin Xiao

  2. Laboratoire de Mathématiques et Physique, University of Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860, Perpignan, France

    Mircea Sofonea

Authors
  1. Mircea Sofonea
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  2. Yi-bin Xiao
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Correspondence to Yi-bin Xiao.

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Communicated by Paolo Vannucci.

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Sofonea, M., Xiao, Yb. Well-Posedness of Minimization Problems in Contact Mechanics. J Optim Theory Appl 188, 650–672 (2021). https://doi.org/10.1007/s10957-020-01801-y

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