Regularity of Minimizers of Shape Optimization Problems involving Perimeter

International audienceWe prove existence and regularity of optimal shapes for the problem$$\min\Big\{P(\Omega)+\mathcal{G}(\Omega):\ \Omega\subset D,\ |\Omega|=m\Big\},$$where $P$ denotes the perimeter, $|\cdot|$ is the volume, and the functional $\mathcal{G}$ is either one of the following: the Dirichlet energy $E_f$, with respect to a (possibly sign-changing) function $f\in L^p$;a spectral functional of the form $F(\lambda_{1},\dots,\lambda_{k})$, where $\lambda_k$ is the $k$th eigenvalue of the Dirichlet Laplacian and $F:\mathbb{R}^k\to\mathbb{R}$ is Lipschitz continuous and increasing in each variable.The domain $D$ is the whole space $\mathbb{R}^d$ or a bounded domain. We also give general assumptions on the functional $\mathcal{G}$ so that the result remains valid