**abstract:**
In this talk I will discuss the regularity theory of shape optimization problems under convexity constraint. More precisely, we are interested in shapes minimizing -among all convex shapes, or only those of given volume- an energy of the form $P(\Omega)+R(\Omega)$ where $P$ is the perimeter of the shape $\Omega$, and $R$ is a perturbation term. This leads us to a notion of quasi-minimizer of the perimeter under convexity constraint, and we show that optimal shapes are $C^{{1,1}$} and that this regularity is optimal.
We note that the perturbation term can include PDE functionals, for example the Dirichlet eigenvalues of the considered shape. We will conclude with perspectives on applications to stability in shape optimization. This is a joint work with Raphaël Prunier

Wed 8 Sep, 14:00 - 14:50, Aula Dini

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