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We consider shape optimization problems for general integral functionals of the calculus of variations, defined on a domain Ω 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.

Abstract-In the present paper we evaluate a number of key Eulerian integrals involving the H-function of several complex variables. Our general Eulerian integral formulas are shown to provide the key formulae from which numerous other potentially useful results for various families of generalized hypergeometric functions of several variables can be derived. Few particular cases are also considered.

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