Structure of shape derivatives for nonsmooth domains

Abstract The object of this paper is to study the Shape gradient and the Shape Hessian by the Velocity (Speed) Method for arbitrary domains with or without constraints. It makes the connection between methods using a family of transformations such as first or second order Perturbations of the Identity Operator. New definitions for Shape derivatives are given. They naturally extend existing theories for Ck or Lipchitzian domains to arbitrary domains without any smoothness conditions on their geometric boundary. In this new framework extensions of the classical structure theorems are given for the Shape gradient and the Shape Hessian.

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