An abstract Lagrangian framework for computing shape derivatives

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