Regularity of Minimizers of Shape Optimization Problems involving Perimeter

We prove existence and regularity of optimal shapes for the problem$\min\Big\{P(Ω)+\mathcal{G}(Ω):\ Ω\subset D,\ |Ω|=m\Big\},$where $P$ denotes the perimeter, $|\cdot|$ is the volume, and the functional $\mathcal{G}$ is either one of the following:\textless{}ul\textgreater{}\textless{}li\textgreater{} the Dirichlet energy $E\_f$, with respect to a (possibly sign-changing) function $f\in L^p$;\textless{}/li\textgreater{}\textless{}li\textgreater{}a spectral functional of the form $F(λ\_{1},\dots,λ\_{k})$, where $λ\_k$ 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.\textless{}/li\textgreater{}\textless{}/ul\textgreater{}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.

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