Regularity of Minimizers of Shape Optimization Problems involving Perimeter

Abstract

We prove existence and regularity of optimal shapes for the problem$$\min \{P(\Omega )+\mathcal{G}(\Omega ):\Omega \subset D,|\Omega |=m\},$$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 $F(\lambda \_1,\dots ,\lambda \_k)$, 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.\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.

Journal de Mathématiques Pures et Appliquées, Elsevier, 2016