Intrinsic expression of the derivatives in domain optimization problems

It is well-known in domain optimization that the derivative of a cost function with respect to the shape of a domain is a distribution with support lying in the boundary of the domain to be optimized. This existence result is completed with a constructive method giving an explicit expression of this distribution, which can be applied in general cases. The relationship between material and local derivates of any order, provided they are well-defined, is also investigated.