{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:18:52Z","timestamp":1787321932186,"version":"build-2736575974"},"reference-count":21,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2015,1]]},"abstract":"<jats:p>The object of this paper is the computation of the domain or boundary expression of a state constrained shape function without explicitly resorting to the material derivative. Our main theorem bypasses the restrictive saddle point assumption in existing differentiability theorems of minimax by introducing the notion of the averaged adjoint state, thus extending the use of minimax theorems to some classes of nonlinear state equations. As an illustration, the theorem is applied to a shape function that depends on a quasi-linear transmission problem. Using a Gagliardo penalization the existence of optimal shapes is established.<\/jats:p>","DOI":"10.1137\/130930807","type":"journal-article","created":{"date-parts":[[2015,7,30]],"date-time":"2015-07-30T15:10:23Z","timestamp":1438269023000},"page":"2017-2039","source":"Crossref","is-referenced-by-count":58,"title":["Minimax Lagrangian Approach to the Differentiability of Nonlinear PDE Constrained Shape Functions Without Saddle Point Assumption"],"prefix":"10.1137","volume":"53","author":[{"given":"Kevin","family":"Sturm","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2015,7,30]]},"reference":[{"key":"atypb1","unstructured":"R. A. Adams and J. J. F. Fournier,\n                      Sobolev Spaces\n                      , 2nd ed., Pure Appl. Math. 140, Elsevier\/Academic Press, Amsterdam, 2003."},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1080\/01630560701381005"},{"key":"atypb3","doi-asserted-by":"crossref","unstructured":"R. Bartle,\n                      A Modern Theory of Integration\n                      , Grad. Stud. Math. 32, AMS, Providence, RI, 2001.","DOI":"10.1090\/gsm\/032"},{"key":"atypb4","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1051\/m2an\/1986200303711","volume":"20","author":"Ce\u0301a J.","year":"1986","journal-title":"Model. Numer. Anal."},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/100810800"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/0362-546X(85)90049-5"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1002\/oca.4660130107"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1137\/0326048"},{"key":"atypb9","doi-asserted-by":"crossref","unstructured":"M. C. Delfour and J.P. Zole\u0301sio,\n                      Shapes and Geometries\n                      , Adv. Des. Control 22, 2nd ed., SIAM, Philadelphia, 2011.","DOI":"10.1137\/1.9780898719826"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1016\/j.bulsci.2011.12.004"},{"key":"atypb11","unstructured":"I. Ekeland and R. Temam,\n                      Convex Analysis and Variational Problems\n                      , North-Holland, Amsterdam, 1976."},{"key":"atypb12","unstructured":"D. H. Fremlin,\n                      Measure Theory\n                      , Vol. 3, Torres Fremlin, Colchester, England, 2004."},{"key":"atypb13","doi-asserted-by":"crossref","unstructured":"A. Henrot and M. Pierre,\n                      Variation et Optimisation de Formes\n                      , Springer, Berlin, 2005.","DOI":"10.1007\/3-540-37689-5"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1051\/cocv:2008002"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-013-0520-4"},{"key":"atypb16","first-page":"989","volume":"14","author":"Kasumba H.","year":"2011","journal-title":"Control Cybernet."},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2005.07.005"},{"key":"atypb18","doi-asserted-by":"crossref","unstructured":"P. Plotnikov and J. Soko\u0142owski,\n                      Compressible Navier-Stokes Equations\n                      , Monogr. Mat. (N.S.) 73, Birkha\u0308user\/Springer Basel AG, Basel, 2012.","DOI":"10.1007\/978-3-0348-0367-0"},{"key":"atypb19","doi-asserted-by":"crossref","unstructured":"J. Soko\u0142owski and J.P. Zole\u0301sio,\n                      Introduction to Shape Optimization\n                      , Springer Ser. Comput. Math. 16, Springer, Berlin, 1992.","DOI":"10.1007\/978-3-642-58106-9_1"},{"key":"atypb20","doi-asserted-by":"crossref","unstructured":"W. P. Ziemer,\n                      Weakly Differentiable Functions\n                      , Grad. Texts Math. 120, Springer, New York, 1989.","DOI":"10.1007\/978-1-4612-1015-3"},{"key":"atypb21","unstructured":"J.P. Zole\u0301sio,\n                      Identification de domains par deformations\n                      , The\u0300se de Doctorate d'e\u0301tat, Universite\u0301 de Nice, Nice, France, 1979."}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/130930807","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:16:20Z","timestamp":1787318180000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/130930807"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,1]]},"references-count":21,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2015,1]]}},"alternative-id":["10.1137\/130930807"],"URL":"https:\/\/doi.org\/10.1137\/130930807","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,1]]}}}