*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.

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