Calculus of Variations and Geometric Measure Theory
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J. Lamboley - A. Novruzi

Polygon as optimal shapes with convexity constraint

created by lamboley on 26 May 2016


Published Paper

Inserted: 26 may 2016
Last Updated: 26 may 2016

Journal: SIAM Control and Optimization
Volume: 48
Number: 5
Pages: 3003-3025
Year: 2009


In this paper, we focus on the following general shape optimization problem: \[ \min\{J(\Omega),\ \Omega\ convex,\ \Omega\in\mathcal S_{ad}\}, \] where $\mathcal S_{ad}$ is a set of 2-dimensional admissible shapes and $J:\mathcal{S}_{ad}\rightarrow\mathbb{R}$ is a shape functional. Using a specific parameterization of the set of convex domains, we derive some extremality conditions (first and second order) for this kind of problem. Moreover, we use these optimality conditions to prove that, for a large class of functionals (satisfying a concavity like property), any solution to this shape optimization problem is a polygon.


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