{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T05:40:34Z","timestamp":1776663634269,"version":"3.51.2"},"reference-count":35,"publisher":"American Mathematical Society (AMS)","issue":"11","license":[{"start":{"date-parts":[[2020,7,30]],"date-time":"2020-07-30T00:00:00Z","timestamp":1596067200000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100004052","name":"King Abdullah University of Science and Technology","doi-asserted-by":"publisher","award":["OSR-CRG2017-3452"],"award-info":[{"award-number":["OSR-CRG2017-3452"]}],"id":[{"id":"10.13039\/501100004052","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004052","name":"King Abdullah University of Science and Technology","doi-asserted-by":"publisher","award":["OSR-CRG2017-3452"],"award-info":[{"award-number":["OSR-CRG2017-3452"]}],"id":[{"id":"10.13039\/501100004052","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Proc. Amer. Math. Soc."],"abstract":"<p>In this paper, we study first-order stationary monotone mean-field games (MFGs) with Dirichlet boundary conditions. Whereas Dirichlet conditions may not be satisfied for Hamilton\u2013Jacobi equations, here we establish the existence of solutions to MFGs that satisfy those conditions. To construct these solutions, we introduce a monotone regularized problem. Applying Schaefer\u2019s fixed-point theorem and using the monotonicity of the MFG, we verify that there exists a unique weak solution to the regularized problem. Finally, we take the limit of the solutions of the regularized problem and, using Minty\u2019s method, we show the existence of weak solutions to the original MFG.<\/p>","DOI":"10.1090\/proc\/14475","type":"journal-article","created":{"date-parts":[[2019,7,24]],"date-time":"2019-07-24T09:20:51Z","timestamp":1563960051000},"page":"4713-4731","source":"Crossref","is-referenced-by-count":18,"special_numbering":"725","title":["Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions"],"prefix":"10.1090","volume":"147","author":[{"given":"Rita","family":"Ferreira","sequence":"first","affiliation":[]},{"given":"Diogo","family":"Gomes","sequence":"additional","affiliation":[]},{"given":"Teruo","family":"Tada","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2019,7,30]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/978-3-642-36433-4_1","article-title":"Finite difference methods for mean field games","author":"Achdou, Yves","year":"2013","ISBN":"https:\/\/id.crossref.org\/isbn\/9783642364327"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"473","DOI":"10.2140\/involve.2017.10.473","article-title":"Existence of positive solutions for an approximation of stationary mean-field games","volume":"10","author":"Almayouf, Nojood","year":"2017","journal-title":"Involve","ISSN":"https:\/\/id.crossref.org\/issn\/1944-4176","issn-type":"print"},{"issue":"4","key":"3","doi-asserted-by":"publisher","first-page":"657","DOI":"10.1007\/s13235-016-0203-5","article-title":"Two numerical approaches to stationary mean-field games","volume":"7","author":"Almulla, Noha","year":"2017","journal-title":"Dyn. 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