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### Abstract

We consider shape optimization problems for general integral functionals of the calculus of variations, defined on a domain $\Omega $ that varies over all subdomains of a given bounded domain $D$ of ${\mathbb{R}}^{d}$. We show in a rather elementary way the existence of a solution that is in general a quasi open set. Under very mild conditions we show that the optimal domain is actually open and with finite perimeter. Some counterexamples show that in general this does not occur.

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