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dc.contributor.authorLamboley, Jimmy
dc.contributor.authorPierre, Michel
dc.contributor.authorNovruzi, Arian
dc.subjectShape optimizationen
dc.subjectconvexity constrainten
dc.subjectoptimality conditionsen
dc.subjectregularity of free boundaryen
dc.titleRegularity and singularities of Optimal Convex shapes in the planeen
dc.typeArticle accepté pour publication ou publié
dc.contributor.editoruniversityotherInstitut de Recherche Mathématique de Rennes (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.editoruniversityotherDepartment of Mathematics and Statistics University of Ottawa;Canada
dc.description.abstractenWe 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.isversionofjnlnameArchive for Rational Mechanics and Analysis

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