Anatomy of the shape Hessian via lie brackets

The goal of this paper is to study the anatomy of the shape Hessian for some classes of smooth shape functionals. A structure theorem gives a decomposition of the shape Hessian in three additive bilinear forms acting on the two fields: the first one acting on the normal components at the boundary, the second one being symmetrical and the third one involving a half of the Lie bracket of the pair of fields at which the shape Hessian is computed. Applications to the commutation of the mixed derivatives and the symmetry of the linear operator which appears in the structure theorem are given.