Shape Optimisation Problems for Integral Functionals and Regularity Properties of Optimal Domain

We consider shape optimization problems for general integral func- tionals of the calculus of variations, defined on a domain Ω that varies over all subdomains of a given bounded domain D of Rd. 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.We also look into further regularity issues posing the shape optimisation problem as a free boundary problem.