Regularity of minimizers of shape optimization problems involving perimeter

We prove the existence and regularity of optimal shapes for the problem min⁡{P(Ω)+G(Ω):Ω⊂D,|Ω|=m}, where P denotes the perimeter, |⋅| is the volume, and the functional G is either one of the following: • the Dirichlet energy Ef, with respect to a (possibly sign-changing) function f∈Lp;• a spectral functional of the form F(λ1,…,λk), where λk is the kth eigenvalue of the Dirichlet Laplacian and F:Rk→R is locally Lipschitz continuous and increasing in each variable.The domain D is the whole space Rd or a bounded domain. We also give general assumptions on the functional G so that the result remains valid.