Some Optimal Design Problems with Perimeter Penalisation

We present measure representation and integral representation results for some integral functionals arising in the context of optimal design problems under certain growth conditions and in the presence of a perimeter penalisation term. The functionals in question correspond to the relaxation with respect to a pair \((\chi ,u)\) , where \(\chi \) is the characteristic function of a set of finite perimeter and u belongs to a suitable function space.