{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T15:33:12Z","timestamp":1787239992594,"version":"build-2736575974"},"reference-count":10,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2015,1]]},"abstract":"<jats:p>This paper deals with the isotropic realizability of a given regular divergence free field $j$ in $\\mathbb{R}^3$ as a current field, namely, to know when $j$ can be written as $\\sigma\\nabla u$ for some isotropic conductivity $\\sigma&gt;0$ and some gradient field $\\nabla u$. The local isotropic realizability in $\\mathbb{R}^3$ is obtained by Frobenius' theorem provided that $j$ and $\\mbox{curl}\\,j$ are orthogonal in $\\mathbb{R}^3$. A counterexample shows that Frobenius' condition is not sufficient to derive the global isotropic realizability in $\\mathbb{R}^3$. However, assuming that $(j,\\mbox{curl}\\,j,j\\times\\mbox{curl}\\,j)$ is an orthogonal basis of $\\mathbb{R}^3$, an admissible conductivity $\\sigma$ is constructed from a combination of the three dynamical flows along the directions $j\/|j|$, $\\mbox{curl}\\,j\/|\\mbox{curl}\\,j|,$ and $(j\/|j|^2)\\times\\mbox{curl}\\,j$. When the field $j$ is periodic, the isotropic realizability in the torus needs in addition a boundedness assumption satisfied by the flow along the third direction $(j\/|j|^2)\\times\\mbox{curl}\\,j$. Several examples illustrate the sharpness of the realizability conditions.<\/jats:p>","DOI":"10.1137\/140989121","type":"journal-article","created":{"date-parts":[[2015,6,24]],"date-time":"2015-06-24T11:32:18Z","timestamp":1435145538000},"page":"1165-1188","source":"Crossref","is-referenced-by-count":2,"title":["Isotropic Realizability of Current Fields in $\\mathbb{R}^3$"],"prefix":"10.1137","volume":"14","author":[{"given":"M.","family":"Briane","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"G. W.","family":"Milton","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2015,6,24]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/PL00004242"},{"key":"atypb2","unstructured":"V. I. Arnol'd,\n                      Ordinary Differential Equations\n                      , translated from the third Russian edition by R. Cooke, Springer Textbook, Springer-Verlag, Berlin, 1992."},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1007\/BF01654133"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2013109"},{"key":"atypb5","unstructured":"H. Cartan,\n                      Calcul Diffe\u0301rentiel\n                      , Hermann, Paris, 1967."},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2011.05.024"},{"key":"atypb7","unstructured":"G. W. Milton,\n                      The Theory of Composites\n                      , Cambridge Monogr. Appl. Comput. Math. 6, Cambridge University Press, Cambridge, UK, 2002."},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1017\/S0308210500030481"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1126\/science.1125907"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1364\/OE.14.009794"}],"container-title":["SIAM Journal on Applied Dynamical Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/140989121","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T15:07:23Z","timestamp":1787238443000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/140989121"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,1]]},"references-count":10,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2015,1]]}},"alternative-id":["10.1137\/140989121"],"URL":"https:\/\/doi.org\/10.1137\/140989121","relation":{},"ISSN":["1536-0040"],"issn-type":[{"value":"1536-0040","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,1]]}}}