Shape optimization problems in a box

Abstract

In this chapter we define two different variational convergences on the family of domains contained in a given box. The term box is widely used in the shape optimization and classically refers to a bounded open set in ℝ^d. The theory of the weak-γ and the strong-γ -convergence¹ of sets in a box was developed in the Euclidean space (see, for example, [21] and the references therein). Nevertheless, as it was shown in [37], this is a theory that uses a purely variational techniques and it can be adapted to a much more general (non-linear) settings as those of measured metric spaces.