Fixed domain approaches in shape optimization problems

This work is a review of results in the approximation of optimal design problems, defined in variable/unknown domains, based on associated optimization problems defined in a fixed 'hold-all' domain, including the family of all admissible open sets. The literature in this respect is very rich and we concentrate on three main approaches: penalization-regularization, finite element discretization on a fixed grid, controllability and control properties of elliptic systems. Comparison with other fixed domain approaches or, in general, with other methods in shape optimization is performed as well and several numerical examples are included.