Optimal shapes for general integral functionals
Annales Henri Lebesgue, Volume 3 (2020) , pp. 261-272.

Keywordsshape optimization, quasi open sets, finite perimeter, integral functionals.

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 ${ℝ}^{d}$. 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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