Regularity and singularities of Optimal Convex shapes in the plane

We focus here on the analysis of the regularity or singularity of solutions Ω0 to shape optimization problems among convex planar sets, namely: J(Ω0) = min{J(Ω), Ω convex, Ω ∈ Sad}, where Sad is a set of 2-dimensional admissible shapes and J : Sad → 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 Ω0 is a W -set, p ∈ [1,∞]. This result applies, for instance, with p = ∞ when the shape functional can be written as J(Ω) = R(Ω) + P (Ω), where R(Ω) = F (|Ω|, Ef (Ω), λ1(Ω)) involves the area |Ω|, the Dirichlet energy Ef (Ω) or the first eigenvalue of the Laplace-Dirichlet operator λ1(Ω), and P (Ω) is the perimeter of Ω, ii) under a suitable concavity assumption on the functional J , we prove that Ω0 is a polygon. This result applies, for instance, when the functional is now written as J(Ω) = R(Ω)−P (Ω), with the same notations as above.