ABSTRACT
In this chapter, we state similar results as the previous chapter but with a method based directly on the Lagrangian flows which do not define a linear space, but has the advantage to first avoid the introduction of transverse fields and could prove appropriate to deal with fluid-solid interaction problems. The chapter is organised as follows,
1. In section 6.2, we introduce the general framework of non-cylindrical Lagrangian shape derivative and we state the structure of the gradient for general functionals under or without state constraints. We also establish an equivalence result between Eulerian and Lagrangian derivatives.
