dc.contributor.author Lamboley, Jimmy HAL ID: 6598 dc.contributor.author Pierre, Michel HAL ID: 6510 dc.contributor.author Novruzi, Arian dc.date.accessioned 2012-01-03T16:06:43Z dc.date.available 2012-01-03T16:06:43Z dc.date.issued 2012 dc.identifier.uri https://basepub.dauphine.fr/handle/123456789/7823 dc.language.iso en en dc.subject Shape optimization en dc.subject convexity constraint en dc.subject optimality conditions en dc.subject regularity of free boundary en dc.subject.ddc 516 en dc.title Regularity and singularities of Optimal Convex shapes in the plane en dc.type Article accepté pour publication ou publié dc.contributor.editoruniversityother Institut de Recherche Mathématique de Rennes (IRMAR) http://www.math.univ-rennes1.fr/irmar CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées de Rennes – Université Rennes 2;France dc.contributor.editoruniversityother Department of Mathematics and Statistics University of Ottawa;Canada dc.description.abstracten We focus here on the analysis of the regularity or singularity of solutions $\Om_{0}$ to shape optimization problems among convex planar sets, namely: $$J(\Om_{0})=\min\{J(\Om),\ \Om\ \textrm{convex},\ \Omega\in\mathcal S_{ad}\},$$ where $\mathcal S_{ad}$ is a set of 2-dimensional admissible shapes and $J:\mathcal{S}_{ad}\rightarrow\R$ is a shape functional. Our main goal is to obtain qualitative properties of these optimal shapes by using first and second order optimality conditions, including the infinite dimensional Lagrange multiplier due to the convexity constraint. We prove two types of results: \begin{itemize} \item[i)] under a suitable convexity property of the functional $J$, we prove that $\Omega_0$ is a $W^{2,p}$-set, $p\in[1,\infty]$. This result applies, for instance, with $p=\infty$ when the shape functional can be written as $J(\Omega)=R(\Omega)+P(\Omega),$ where $R(\Om)=F(|\Om|,E_{f}(\Om),\la_{1}(\Om))$ involves the area $|\Omega|$, the Dirichlet energy $E_f(\Omega)$ or the first eigenvalue of the Laplace-Dirichlet operator $\lambda_1(\Omega)$, and $P(\Omega)$ is the perimeter of $\Om$, \item[ii)] under a suitable concavity assumption on the functional $J$, we prove that $\Omega_{0}$ is a polygon. This result applies, for instance, when the functional is now written as $J(\Om)=R(\Omega)-P(\Omega),$ with the same notations as above. \end{itemize} en dc.relation.isversionofjnlname Archive for Rational Mechanics and Analysis dc.relation.isversionofjnlvol 205 dc.relation.isversionofjnlissue 1 dc.relation.isversionofjnldate 2012 dc.relation.isversionofjnlpages 313-343 dc.relation.isversionofdoi http://dx.doi.org/10.1007/s00205-012-0514-7 dc.identifier.urlsite http://hal.archives-ouvertes.fr/hal-00651557/fr/ en dc.description.sponsorshipprivate oui en dc.relation.isversionofjnlpublisher Springer dc.subject.ddclabel Géométrie en
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