# Regularity of minimizers of shape optimization problems involving perimeter

created by dephilipp on 20 May 2016
modified on 17 Nov 2016

[BibTeX]

Accepted Paper

Inserted: 20 may 2016
Last Updated: 17 nov 2016

Journal: J. Math. Pures Appl.
Year: 2016

Abstract:

We 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:

1) The Dirichlet energy $E_f$, with respect to a (possibly sign-changing) function $f\in L^p$;

2) 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.