*Published Paper*

**Inserted:** 20 may 2016

**Last Updated:** 21 apr 2018

**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: 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.

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