{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T23:50:39Z","timestamp":1774396239981,"version":"3.50.1"},"reference-count":30,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,10,17]],"date-time":"2024-10-17T00:00:00Z","timestamp":1729123200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Union\u2019s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement","award":["823731 CONMECH"],"award-info":[{"award-number":["823731 CONMECH"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We consider an elliptic variational\u2013hemivariational inequality P in a real reflexive Banach space, governed by a set of constraints K. Under appropriate assumptions of the data, this inequality has a unique solution u\u2208K. We associate inequality P to a sequence of elliptic variational\u2013hemivariational inequalities {Pn}, governed by a set of constraints K\u02dc\u2283K, a sequence of parameters {\u03bbn}\u2282R+, and a function \u03c8. We prove that if, for each n\u2208N, the element un\u2208K\u02dc represents a solution to Problem Pn, then the sequence {un} converges to u as \u03bbn\u21920. Based on this general result, we recover convergence results for various associated penalty methods previously obtained in the literature. These convergence results are obtained by considering particular choices of the set K\u02dc and the function \u03c8. The corresponding penalty methods can be applied in the study of various inequality problems. To provide an example, we consider a purely hemivariational inequality that describes the equilibrium of an elastic membrane in contact with an obstacle, the so-called foundation.<\/jats:p>","DOI":"10.3390\/axioms13100721","type":"journal-article","created":{"date-parts":[[2024,10,17]],"date-time":"2024-10-17T09:14:03Z","timestamp":1729156443000},"page":"721","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Penalty Method for Elliptic Variational\u2013Hemivariational Inequalities"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6110-1433","authenticated-orcid":false,"given":"Mircea","family":"Sofonea","sequence":"first","affiliation":[{"name":"Laboratoire de Math\u00e9matiques et Physique, University of Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2813-0419","authenticated-orcid":false,"given":"Domingo A.","family":"Tarzia","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Facultad de Ciencias Empresariales (FCE), Universidad Austral, Paraguay 1950, Rosario S2000FZF, Argentina"},{"name":"Consejo Nacional de Investigaciones Cient\u00edficas y T\u00e9cnicas (CONICET), Rosario S2000EZP, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,17]]},"reference":[{"key":"ref_1","unstructured":"Baiocchi, C., and Capelo, A. 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