Tykhonov well-posedness of a mixed variational problem. (4th March 2022)

• Record Type: Journal Article
• Authors: Cai, Dong-ling; Sofonea, Mircea; Xiao, Yi-bin
• Is Part Of: Optimization. Volume 71:Number 3(2022)
• Page Start: 561
• Page End: 581
• Publication Date: 2022-03-04
• Subjects: Mixed variational problem -- approximating sequence -- Tykhonov well-posedness -- contact problem -- unilateral constraint; 35M86 -- 47J20 -- 49J40 -- 74M15 -- 74G25; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2020.1808646
• Languages: English
• ISSNs: 0233-1934

Abstract

ABSTRACT: We consider a mixed variational problem governed by a nonlinear operator and a set of constraints. Existence, uniqueness and convergence results for this problem have already been obtained in the literature. In this current paper we complete these results by proving the well-posedness of the problem, in the sense of Tykhonov. To this end we introduce a family of approximating problems for which we state and prove various equivalence and convergence results. We illustrate these abstract results in the study of a frictionless contact model with elastic materials. The process is assumed to be static and the contact is with unilateral constraints. We derive a weak formulation of the model which is in the form of a mixed variational problem with unknowns being the displacement field and the Lagrange multiplier. Then, we prove various results on the corresponding mixed problem, including its well-posedness in the sense of Tykhonov, under various assumptions on the data. Finally, we provide mechanical interpretation of our results.