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Regularity and singularities of Optimal Convex shapes in the plane

Lamboley, Jimmy; Pierre, Michel; Novruzi, Arian (2012), Regularity and singularities of Optimal Convex shapes in the plane, Archive for Rational Mechanics and Analysis, 205, 1, p. 313-343. http://dx.doi.org/10.1007/s00205-012-0514-7

Type
Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00651557/fr/
Date
2012
Journal name
Archive for Rational Mechanics and Analysis
Volume
205
Number
1
Publisher
Springer
Pages
313-343
Publication identifier
http://dx.doi.org/10.1007/s00205-012-0514-7
Metadata
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Author(s)
Lamboley, Jimmy
Pierre, Michel
Novruzi, Arian
Abstract (EN)
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}
Subjects / Keywords
Shape optimization; convexity constraint; optimality conditions; regularity of free boundary

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