{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T14:09:49Z","timestamp":1790690989201,"version":"4.1.0"},"reference-count":27,"publisher":"SAGE Publications","license":[{"start":{"date-parts":[[2026,9,27]],"date-time":"2026-09-27T00:00:00Z","timestamp":1790467200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Mathematics and Mechanics of Solids"],"abstract":"<jats:p>We systematically discuss the equivalence of two variational formulations of magnetostatics, in terms of magnetization and magnetic field, on the one hand, and the single-field formulation using only magnetic induction. To demonstrate that this link is stable also when the magnetic laws are coupled with other variational static models, elasticity is included in the models as well. Interestingly, even though the corresponding magnetoelastic energy densities in the material can be computed via Legendre\u2013Fenchel transform in the magnetic state variables, the two formulations are not linked by standard convex duality on the level of the functionals. In addition, convexity and coercivity of the given functional are neither required for the transformation nor always preserved by it.<\/jats:p>","DOI":"10.1177\/10812865261477698","type":"journal-article","created":{"date-parts":[[2026,9,27]],"date-time":"2026-09-27T17:10:22Z","timestamp":1790529022000},"update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":0,"title":["On the single-field formulation in magnetostatics"],"prefix":"10.1177","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2534-2149","authenticated-orcid":false,"given":"Stefan","family":"Kr\u00f6mer","sequence":"first","affiliation":[{"name":"Institute of Information Theory and Automation, The Czech Academy of Sciences, Prague, Czech Republic"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8801-7461","authenticated-orcid":false,"given":"Giuseppe","family":"Tomassetti","sequence":"additional","affiliation":[{"name":"Department of Industrial, Electronic, and Mechanical Engineering, Roma Tre University, Italy"},{"name":"Institute for Applied Mathematics \u201cMauro Picone\u201d (IAC), National Research Council (CNR), Rome, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"179","published-online":{"date-parts":[[2026,9,27]]},"reference":[{"key":"e_1_3_3_2_2","doi-asserted-by":"crossref","unstructured":"William F Brown J. Magnetoelastic interactions Springer tracts in natural philosophy. Vol. 9. Springer-Verlag 1966. https:\/\/doi.org\/10.1007\/978-3-642-87396-6.","DOI":"10.1007\/978-3-642-87396-6"},{"key":"e_1_3_3_3_2","doi-asserted-by":"publisher","DOI":"10.1080\/01418639308226405"},{"key":"e_1_3_3_4_2","volume-title":"Classical electrodynamics","author":"Jackson JD","year":"1999","unstructured":"Jackson JD. Classical electrodynamics. 3rd ed. John Wiley & Sons, 1999.","edition":"3"},{"key":"e_1_3_3_5_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.58.736"},{"key":"e_1_3_3_6_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.60.139"},{"key":"e_1_3_3_7_2","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.17.15"},{"key":"e_1_3_3_8_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.105.1479"},{"key":"e_1_3_3_9_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF00376811"},{"key":"e_1_3_3_10_2","doi-asserted-by":"publisher","DOI":"10.1007\/s002110100286"},{"key":"e_1_3_3_11_2","doi-asserted-by":"crossref","unstructured":"DeSimone A Kohn RV M\u00fcller S et al. Recent analytical developments in micromagnetics. Elsevier 2006. pp.269\u2013381. https:\/\/doi.org\/10.1016\/B978-012480874-4\/50015-4.","DOI":"10.1016\/B978-012480874-4\/50015-4"},{"key":"e_1_3_3_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmps.2004.04.007"},{"key":"e_1_3_3_13_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00707-003-0061-2"},{"key":"e_1_3_3_14_2","doi-asserted-by":"publisher","DOI":"10.1177\/1081286507079832"},{"key":"e_1_3_3_15_2","doi-asserted-by":"publisher","DOI":"10.1177\/1081286517696536"},{"key":"e_1_3_3_16_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmps.2017.04.016"},{"key":"e_1_3_3_17_2","doi-asserted-by":"publisher","DOI":"10.1177\/1081286520975808"},{"key":"e_1_3_3_18_2","doi-asserted-by":"publisher","DOI":"10.1137\/080738179"},{"key":"e_1_3_3_19_2","volume-title":"Convex analysis, Princeton Mathematical Series","author":"Rockafellar RT","year":"1970","unstructured":"Rockafellar RT. Convex analysis, Princeton Mathematical Series. Vol. 28. Princeton University Press, 1970."},{"key":"e_1_3_3_20_2","doi-asserted-by":"crossref","unstructured":"Clarke FH. Optimization and nonsmooth analysis Classics in applied mathematics. Vol. 5. 2nd ed. Society for Industrial and Applied Mathematics (SIAM) 1990. https:\/\/doi.org\/10.1137\/1.9781611971309.","DOI":"10.1137\/1.9781611971309"},{"key":"e_1_3_3_21_2","doi-asserted-by":"publisher","DOI":"10.1051\/cocv\/2014010"},{"key":"e_1_3_3_22_2","doi-asserted-by":"publisher","DOI":"10.1051\/cocv\/2021074"},{"key":"e_1_3_3_23_2","doi-asserted-by":"publisher","DOI":"10.1017\/S0308210507000662"},{"key":"e_1_3_3_24_2","doi-asserted-by":"publisher","DOI":"10.1038\/nmat2126"},{"key":"e_1_3_3_25_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.105.017403"},{"key":"e_1_3_3_26_2","doi-asserted-by":"crossref","unstructured":"Galdi GP. An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems. 2nd ed. Springer 2011. https:\/\/doi.org\/10.1007\/978-0-387-09620-9.","DOI":"10.1007\/978-0-387-09620-9"},{"key":"e_1_3_3_27_2","volume-title":"Conjugate duality and optimization, CBMS-NSF regional conference series in applied mathematics","author":"Rockafellar RT","year":"1974","unstructured":"Rockafellar RT. Conjugate duality and optimization, CBMS-NSF regional conference series in applied mathematics. Vol. 16. Society for Industrial and Applied Mathematics (SIAM), 1974."},{"key":"e_1_3_3_28_2","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2010.28.511"}],"container-title":["Mathematics and Mechanics of Solids"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/10812865261477698","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.1177\/10812865261477698","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/10812865261477698","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T13:52:12Z","timestamp":1790689932000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.1177\/10812865261477698"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,27]]},"references-count":27,"alternative-id":["10.1177\/10812865261477698"],"URL":"https:\/\/doi.org\/10.1177\/10812865261477698","relation":{},"ISSN":["1081-2865","1741-3028"],"issn-type":[{"value":"1081-2865","type":"print"},{"value":"1741-3028","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,27]]},"article-number":"10812865261477698"}}