Tykhonov well-posedness of a mixed variational problem

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.

Keywords:

• Mixed variational problem
• approximating sequence
• Tykhonov well-posedness
• contact problem
• unilateral constraint

2010 Mathematics Subject Classifications:

• 35M86
• 47J20
• 49J40
• 74M15
• 74G25

Acknowledgments

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

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This project has received funding from the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No 823731 CONMECH. This research was also supported by the National Natural Science Foundation of China [grant number 11771067], the Applied Basic Project of Sichuan Province [grant number 2019YJ0204] and the Fundamental Research Funds for the Central Universities [grant number ZYGX2019J095]