Well-Posedness of Minimization Problems in Contact Mechanics

Mircea Sofonea () and Yi-bin Xiao ()
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Mircea Sofonea: University of Electronic Science and Technology of China
Yi-bin Xiao: University of Electronic Science and Technology of China

Journal of Optimization Theory and Applications, 2021, vol. 188, issue 3, No 3, 650-672

Abstract: 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.

Keywords: Minimization problem; Tykhonov well-posedness; Contact problem; Elastic material; Convergence results; 49J45; 90C48; 90J31 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10957-020-01801-y

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