{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:42:34Z","timestamp":1787323354244,"version":"build-2736575974"},"reference-count":49,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"name":"IMATI - C.N.R."},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["M 2876"],"award-info":[{"award-number":["M 2876"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100012740","name":"Gruppo Nazionale per l'Analisi Matematica, la Probabilit\u00e0 e le loro Applicazioni","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100012740","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["21K03309"],"award-info":[{"award-number":["21K03309"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003407","name":"Ministero dell'Istruzione, dell'Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100003407","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100008608","name":"Sumitomo Foundation","doi-asserted-by":"publisher","award":["190367"],"award-info":[{"award-number":["190367"]}],"id":[{"id":"10.13039\/100008608","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Math. Anal."],"published-print":{"date-parts":[[2022,6]]},"abstract":"<jats:p>An asymptotic analysis for a system with an equation and a dynamic boundary condition of Cahn--Hilliard type is carried out as the coefficient of the surface diffusion acting on the phase variable tends to 0, thus obtaining a forward-backward dynamic boundary condition at the limit. This is done in a very general setting, with nonlinear terms admitting maximal monotone graphs both in the bulk and on the boundary. The two graphs are related by a growth condition with the boundary graph that dominates the other one. It turns out that in the limiting procedure the solution of the problem loses some regularity and the limit equation has to be properly interpreted in the sense of a subdifferential inclusion. However, the limit problem is still well-posed since a continuous dependence estimate can be proved. Moreover, in the case when the two graphs exhibit the same growth, it is shown that the solution enjoys more regularity and the boundary condition holds almost everywhere. An error estimate can also be shown for a suitable order of the diffusion parameter.<\/jats:p>","DOI":"10.1137\/21m142441x","type":"journal-article","created":{"date-parts":[[2022,6,2]],"date-time":"2022-06-02T10:59:20Z","timestamp":1654167560000},"page":"3292-3315","source":"Crossref","is-referenced-by-count":10,"title":["The Cahn--Hilliard Equation with Forward-Backward Dynamic Boundary Condition via Vanishing Viscosity"],"prefix":"10.1137","volume":"54","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7921-5041","authenticated-orcid":true,"given":"Pierluigi","family":"Colli","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Takeshi","family":"Fukao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6928-8944","authenticated-orcid":true,"given":"Luca","family":"Scarpa","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2022,6,2]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-5542-5"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202517500129"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.3934\/cpaa.2018049"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1007\/s00526-020-1715-9"},{"key":"atypb5","volume-title":"North-Holland","author":"Br\u00e9zis H.","year":"1973"},{"key":"atypb6","first-page":"1.1","volume-title":"Finite Element Handbook","author":"Brezzi F.","year":"1987"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1016\/0001-6160(61)90182-1"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1063\/1.1744102"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2012.11.010"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2015.07.011"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2016.01.032"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1515\/anona-2018-0055"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1007\/s00030-020-00654-8"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1002\/mana.201900361"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2017.03.020"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2014.05.008"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1515\/anona-2015-0035"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1137\/16M1087539"},{"key":"atypb19","first-page":"5","volume":"10","author":"Colli P.","year":"2017","journal-title":"Vestn. Yuzhno-Ural. Gos. Univ. Ser. Mat. Model. Program."},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1007\/s10231-018-0732-1"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1137\/17M1146786"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1007\/s41808-018-0021-6"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.3233\/ASY-161380"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1080\/00207179.2019.1680870"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.3233\/ASY-201646"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.3934\/cpaa.2017093"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1002\/mma.757"},{"key":"atypb28","first-page":"1241","volume":"12","author":"Gal C. G.","year":"2007","journal-title":"Adv. Differential Equations"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1137\/19M1258840"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2021.112619"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.3934\/cpaa.2009.8.881"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2010.12.007"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6544\/ab8351"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2020090"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-019-01356-x"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.3934\/Math.2017.2.479"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1002\/mma.464"},{"key":"atypb38","first-page":"329","volume-title":"Material Instabilities in Continuum Mechanics (Edinburgh","author":"Novick-Cohen A.","year":"1985"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1090\/qam\/973383"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1991-1015926-7"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.3934\/cpaa.2021181"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2018.10.007"},{"key":"atypb43","doi-asserted-by":"crossref","first-page":"83","DOI":"10.57262\/ade\/1355926869","volume":"8","author":"Racke R.","year":"2003","journal-title":"Adv. Differential Equations"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2018.09.034"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-020-09652-9"},{"key":"atypb46","doi-asserted-by":"publisher","DOI":"10.1007\/BF01762360"},{"key":"atypb47","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2014.06.036"},{"key":"atypb48","doi-asserted-by":"crossref","first-page":"1519","DOI":"10.3934\/dcdss.2017078","volume":"10","author":"Tomassetti G.","year":"2017","journal-title":"Discrete Contin. Dyn. Syst. Ser. S"},{"key":"atypb49","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2004.05.004"}],"container-title":["SIAM Journal on Mathematical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/21M142441X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:57:29Z","timestamp":1787320649000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/21M142441X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6]]},"references-count":49,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2022,6]]}},"alternative-id":["10.1137\/21M142441X"],"URL":"https:\/\/doi.org\/10.1137\/21m142441x","relation":{},"ISSN":["0036-1410","1095-7154"],"issn-type":[{"value":"0036-1410","type":"print"},{"value":"1095-7154","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,6]]}}}