A convergence criterion for elliptic variational inequalities

Abstract

AbstractWe 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 {un}⊂X which guarantee the convergence un→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 inequalityconvergence criterionconvergence resultswell-posednesscontactheat transferunilateral constraint2010 MSC: 47J2049J4040A0574M1574M1035J20 AcknowledgmentsThis 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 statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work was supported by European Commission[].