Structure of shape derivatives

Abstract.

In this paper, we describe the precise structure of second "shape derivatives", that is derivatives of functions whose argument is a variable subset of $\mathbb{R}^N$. This is done for Fréchet derivatives in adequate Banach spaces. Besides the structure itself, interest lies in the way it is derived: the starting point is a "functional analytic" statement of the well-known fact that small regular perturbations of a given regular domain may be "uniquely" represented through normal deformations of the boundary of this domain. The approach involves the implicit function theorem in a convenient functional space. A consequence of this "normal representation" property is that any shape functional may be described through a functional depending on functions defined only on the boundary of the given domain. Differentiating twice this representation leads to the structure theorem. We recover the fact that, at critical shapes, the second derivative around the given domain depends only on the normal component of the deformation vector-field at its boundary. Some examples are explicitly computed.