A convergence criterion for elliptic variational inequalities

Abstract

We consider an elliptic variational inequality with unilateral constraints in a Hilbert space X which, under appropriate assumptions on the data, has a unique solution u. We formulate a convergence criterion to the solution u, i.e. we provide necessary and sufficient conditions on a sequence $\{u_n\}\subset X$ which guarantee the convergence $u_n\to u$ in the space X. Then we illustrate the use of this criterion to recover well-known convergence results and well-posedness results in the sense of Tykhonov and Levitin–Polyak. We also provide two applications of our results, in the study of a heat transfer problem and an elastic frictionless contact problem, respectively.

Keywords:

• Elliptic variational inequality
• convergence criterion
• convergence results
• well-posedness
• contact
• heat transfer
• unilateral constraint

2010 MSC:

• 47J20
• 49J40
• 40A05
• 74M15
• 74M10
• 35J20

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. The second author was also supported by the Ministry of Science and Higher Education of Republic of Poland under Grant No. 440328/PnH2/2019, and in part from National Science Centre, Poland under project OPUS no. 2021/41/B/ST1/01636.

Disclosure statement

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

Additional information

Funding

This work was supported by European Commission[].