{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:42:59Z","timestamp":1787323379874,"version":"build-2736575974"},"reference-count":28,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"name":"MIUR-PRIN","award":["2020F3NCPX"],"award-info":[{"award-number":["2020F3NCPX"]}]},{"name":"MUR"},{"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\/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"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Math. Anal."],"published-print":{"date-parts":[[2024,8,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>This work investigates the well-posedness and optimal control of a sixth-order Cahn\u2013Hilliard equation, a higher-order variant of the celebrated and well-established Cahn\u2013Hilliard equation. The equation is endowed with a source term, where the control variable enters as a distributed mass regulator. The inclusion of additional spatial derivatives in the sixth-order formulation enables the model to capture curvature effects, leading to a more accurate depiction of isothermal phase separation dynamics in complex materials systems. We provide a well-posedness result for the aforementioned system when the corresponding nonlinearity of double-well shape is regular and then analyze a corresponding optimal control problem. For the latter, existence of optimal controls is established, and the first-order necessary optimality conditions are characterized via a suitable variational inequality. These results aim at contributing to improving the understanding of the mathematical properties and control aspects of the sixth-order Cahn\u2013Hilliard equation, offering potential applications in the design and optimization of materials with tailored microstructures and properties.<\/jats:p>","DOI":"10.1137\/24m1630372","type":"journal-article","created":{"date-parts":[[2024,7,9]],"date-time":"2024-07-09T04:01:56Z","timestamp":1720497716000},"page":"4928-4969","source":"Crossref","is-referenced-by-count":6,"title":["Curvature Effects in Pattern Formation: Well-Posedness and Optimal Control of a Sixth-Order Cahn\u2013Hilliard Equation"],"prefix":"10.1137","volume":"56","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7921-5041","authenticated-orcid":true,"given":"Pierluigi","family":"Colli","sequence":"first","affiliation":[{"name":"Dipartimento di Matematica \u201cF. Casorati,\u201d Universit\u00e0 di Pavia, and Research Associate at the IMATI \u2013 C.N.R. Pavia, I-27100 Pavia, Italy."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gianni","family":"Gilardi","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica \u201cF. Casorati,\u201d Universit\u00e0 di Pavia, and Research Associate at the IMATI \u2013 C.N.R. Pavia, I-27100 Pavia, Italy."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7025-977X","authenticated-orcid":true,"given":"Andrea","family":"Signori","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica, Politecnico di Milano, I-20133 Milano, Italy, Alexander von Humboldt Research Fellow."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J\u00fcrgen","family":"Sprekels","sequence":"additional","affiliation":[{"name":"Weierstrass Institute for Applied Analysis and Stochastics, D-10117 Berlin, Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2024,7,9]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2008.v6.n2.a3"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1063\/1.1744102"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6544\/ace2a7"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.4171\/ifb\/261"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1137\/120874126"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1137\/130941432"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1137\/13092705X"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/18\/3\/016"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2004.01.029"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.3934\/cpaa.2009.8.881"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.65.1116"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202514500365"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2015.35.2539"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1137\/100817590"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1137\/110834123"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1080\/00036811.2014.972384"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1007\/s13398-015-0227-5"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1002\/mma.464"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.3934\/cpaa.2011.10.1823"},{"key":"ref20","doi-asserted-by":"crossref","first-page":"517","DOI":"10.3934\/dcdss.2013.6.517","volume":"6","author":"Paw\u0142ow I.","year":"2013","journal-title":"Discrete Contin. 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