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Regularity and singularities of Optimal Convex shapes in the plane

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Date
2012
Link to item file
http://hal.archives-ouvertes.fr/hal-00651557/fr/
Dewey
Géométrie
Sujet
Shape optimization; convexity constraint; optimality conditions; regularity of free boundary
Journal issue
Archive for Rational Mechanics and Analysis
Volume
205
Number
1
Publication date
2012
Article pages
313-343
Publisher
Springer
DOI
http://dx.doi.org/10.1007/s00205-012-0514-7
URI
https://basepub.dauphine.fr/handle/123456789/7823
Collections
  • CEREMADE : Publications
Metadata
Show full item record
Author
Lamboley, Jimmy
Pierre, Michel
Novruzi, Arian
Type
Article accepté pour publication ou publié
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}

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