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Regularity and singularities of Optimal Convex shapes in the plane

We focus here on the analysis of the regularity or singularity of solutions $\Om_{0}$ to shape optimization problems among convex planar sets, namely: $$ J(\Om_{0})=\min\{J(\Om),\ \Om\ \textrm{convex},\ Ω\in\mathcal S_{ad}\}, $$ 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: i) under a suitable convexity property of the functional $J$, we prove that $Ω_0$ 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(Ω)=R(Ω)+P(Ω), $ where $R(\Om)=F(|\Om|,E_{f}(\Om),\la_{1}(\Om))$ involves the area $|Ω|$, the Dirichlet energy $E_f(Ω)$ or the first eigenvalue of the Laplace-Dirichlet operator $λ_1(Ω)$, and $P(Ω)$ is the perimeter of $\Om$, ii) under a suitable concavity assumption on the functional $J$, we prove that $Ω_{0}$ is a polygon. This result applies, for instance, when the functional is now written as $ J(\Om)=R(Ω)-P(Ω), $ with the same notations as above.