Do Optimal Shapes Exist?

This paper surveys the talk given by the author at the “Seminario Matematico e Fisico di Milano” in November 2006. It deals with the existence question for shape optimization problems associated to the Dirichlet Laplacian. Existence of solutions is seen from both geometrical and functional (γ-convergence) point of view and is discussed in relationship with the optimality conditions and numerical algorithms. Several examples are given concerning isoperimetric inequalities for eigenvalues and shape control problems. Received: August 2007