  In this paper we study an abstract framework for computing shape derivatives
of functionals subject to PDE constraints. We revisit the Lagrangian approach
using the implicit function theorem in an abstract setting tailored for
applications to shape optimization. This abstract framework yields practical
formulae to compute the derivative of a shape functional, the material
derivative of the state, and the adjoint state. Furthermore, it allows to gain
insight on the duality between the material derivative of the state and the
adjoint state. We show several applications of our main result to the
computation of distributed shape derivatives for problems involving linear
elliptic, nonlinear elliptic, parabolic PDEs and distributions. We compare our
approach with other techniques for computing shape derivatives including the
material derivative method and the averaged adjoint method.
