We consider shape optimization problems for general integral functionals of the calculus of variations, defined on a domain that varies over all subdomains of a given bounded domain of . 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.
[But89] Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Reseqrch Notes in Mathematics Series, Volume 207, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989 | MR 1020296 | Zbl 0669.49005
[Giu03] Direct Methods in the Calculus of Variations, World Scientific Publishing, Singapore, 2003 | Zbl 1028.49001
[HM05] Variation et Optimisation de Formes. Une Analyse Géométrique, Mathématiques & Applications, Volume 48, Springer, 2005 | Zbl 1098.49001
[Zie89] Weakly Differentiable Functions, Springer, 1989 | Zbl 0692.46022