ETD

• finite perimeter
• integral func- tionals.
• quasi open sets
• shape optimization

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.